diff --git a/.github/workflows/python-app.yml b/.github/workflows/python-app.yml
index 2ad4b0f..e581eeb 100644
--- a/.github/workflows/python-app.yml
+++ b/.github/workflows/python-app.yml
@@ -19,10 +19,10 @@ jobs:
 
     steps:
     - uses: actions/checkout@v4
-    - name: Set up Python 3.10
+    - name: Set up Python 3.12.4
       uses: actions/setup-python@v3
       with:
-        python-version: "3.10"
+        python-version: "3.12.4"
     - name: Install dependencies
       run: |
         sudo apt-get update
diff --git a/plutho/mesh/custom_mesh_parser.py b/plutho/mesh/custom_mesh_parser.py
index d8474c3..14095e8 100644
--- a/plutho/mesh/custom_mesh_parser.py
+++ b/plutho/mesh/custom_mesh_parser.py
@@ -10,6 +10,10 @@
 import numpy as np
 import numpy.typing as npt
 
+# Local libraries
+from .gmsh_parser import element_to_nodes_map, \
+    element_to_boundary_nodes_map
+
 
 class TokenType(Enum):
     SectionStart = "SectionStart",
@@ -140,10 +144,14 @@ class CustomParser:
     nodes: npt.NDArray
     elements: List[Element]
 
-    def __init__(self, file_path: str):
+    element_order: int
+
+    def __init__(self, file_path: str, element_order: int):
         if not os.path.isfile(file_path):
             raise IOError(f"Mesh file {file_path} not found.")
 
+        self.element_order = element_order
+
         # Load text
         mesh_text = ""
         with open(file_path, "r", encoding="UTF-8") as fd:
@@ -167,9 +175,10 @@ def get_mesh_nodes_and_elements(self) -> Tuple[npt.NDArray, npt.NDArray]:
             if element.type == 2:
                 triangle_elements.append(element)
 
+        nodes_per_element = element_to_nodes_map[2, self.element_order]
         number_of_triangle_elements = len(triangle_elements)
         elements = np.zeros(
-            shape=(number_of_triangle_elements, 3),
+            shape=(number_of_triangle_elements, nodes_per_element),
             dtype=np.int64
         )
 
@@ -235,11 +244,13 @@ def get_triangle_elements_by_physical_group(
         elements_pg = self.get_elements_by_physical_group(physical_group_name)
         _, elements = self.get_mesh_nodes_and_elements()
 
+        nodes_per_element = element_to_nodes_map[2, self.element_order]
+
         # Find triangle elements, which share the same nodes
         triangle_elements = []
         for element_pg in elements_pg:
             for element in elements:
-                if len(element) != 3:
+                if len(element) != nodes_per_element:
                     continue
 
                 found = True
@@ -391,10 +402,12 @@ def _parse_elements(self):
                 tags.append(int(token.value))
 
             # Parse nodes
-            # element_type = 1 -> Line -> 2 Points
-            # element_type = 2 -> Triangle -> 3 Points
+            number_of_nodes = element_to_nodes_map[
+                element_type,
+                self.element_order
+            ]
             nodes = []
-            for _ in range(element_type+1):
+            for _ in range(number_of_nodes):
                 token = self._next_token()
                 expect_token(token, TokenType.TokenInt)
                 # Subtract 1 because in gmsh the indices start at 1
diff --git a/plutho/mesh/gmsh_parser.py b/plutho/mesh/gmsh_parser.py
index 94c1efa..be69498 100644
--- a/plutho/mesh/gmsh_parser.py
+++ b/plutho/mesh/gmsh_parser.py
@@ -9,11 +9,28 @@
 import numpy as np
 import numpy.typing as npt
 
+
+# First index represents the element type:
+# 1: Line
+# 2: Triangle
+# Second index represents the element order
+element_to_nodes_map = np.array([
+    [-1, -1, -1, -1],
+    [-1, 2, 3, 4],
+    [-1, 3, 6, 10]
+])
+
+# Since its the same for lines and triangles, this map only has one row
+# and the index represents the element order
+element_to_boundary_nodes_map = np.array([-1, 2, 3, 4])
+
+
 class GmshParser:
     """Class to use the gmsh python interface to read the mesh files.
     """
+    element_order: int
 
-    def __init__(self, file_path):
+    def __init__(self, file_path, element_order):
         if not gmsh.is_initialized():
             gmsh.initialize()
 
@@ -22,6 +39,8 @@ def __init__(self, file_path):
         else:
             raise IOError(f"Mesh file {file_path} not found.")
 
+        self.element_order = element_order
+
     @staticmethod
     def _get_nodes(node_tags, node_coords, _=None) -> npt.NDArray:
         """Internal function to return a list of the node coordinates based
@@ -43,36 +62,85 @@ def _get_nodes(node_tags, node_coords, _=None) -> npt.NDArray:
 
         return nodes
 
-    @staticmethod
     def _get_elements(
+        self,
         element_types,
         element_tags,
         element_node_tags
-    ) -> npt.NDArray:
-        """Returns a list of elements used in the simulation from the gmsh
-        getElements() api function.
+    ) -> Dict[int, npt.NDArray]:
+        """
+        For each given element type a list of elements is returned.
 
         Parameters:
-            element_types: Element types from gmsh api. Similar to the element
-                dimension so 2 equals triangles.
-            element_tags: Tags of the element per element type.
-            element_node_tags: Tags of the nodes per element tag.
+            element_types: List of element types for which the elements lists
+                are extracted.
+            element_tags: Tags of the elements per element type.
+            element_node_tags: Tags of the nodes of the elements per element
+                type.
 
         Returns:
-            A list of lists where each inner list contains the node indices for
-            the triangle at the outer list index.
+            A list of elements for each given element type:
+            List of elements [e1, e2, e3, ...] where each element consists of
+            its node tags: e1: [n1, n2, ...].
         """
-        elements = np.zeros(shape=(len(element_tags[1]), 3), dtype=int)
+        elements_per_type = {}
+
+        # Types of the given elements: 2D -> Triangle or 1D -> Line
+        # Iterate over all given types
         for i, element_type in enumerate(element_types):
-            if element_type == 2:
-                # Only looking for 3-node triangle elements
-                current_element_tags = element_tags[i]
-                current_node_tags = element_node_tags[i]
-                for j, _ in enumerate(current_element_tags):
-                    # 1 is subtracted because the indices in gmsh start with 1.
-                    elements[j] = current_node_tags[3*j:3*(j+1)] - np.ones(3)
+            elements = []
+            element_indices = []
+
+            # For each type get the order to calculate the number of nodes
+            # it consists of
+            _, dim, order, _, _, _ = \
+                gmsh.model.mesh.getElementProperties(
+                    element_type
+                )
 
-        return elements
+            match dim:
+                case 0:
+                    nodes_per_element = 1
+                case 1:
+                    # Line element
+                    nodes_per_element = order+1
+                case 2:
+                    # Triangle element
+                    nodes_per_element = int(1/2*(order+1)*(order+2))
+                case _:
+                    raise ValueError("Only supporting 2D meshes")
+
+            # The element tags and element_node_tags lists are nested lists
+            # which contain the element tags and node tags of the elements of
+            # the current type at the respective index
+            current_element_tags = element_tags[i]
+            current_node_tags = element_node_tags[i]
+
+            # Now iterate over every element of the current type and extract
+            # the node indices
+            for j, _ in enumerate(current_element_tags):
+                # 1 is subtracted because the indices in gmsh start with 1.
+                elements.append(current_node_tags[
+                        nodes_per_element*j:nodes_per_element*(j+1)
+                    ] - np.ones(nodes_per_element)
+                )
+                element_indices.append(j)
+
+            if len(elements) == 0:
+                raise ValueError(
+                    "Couldn't return elements because the list is empty."
+                )
+
+            elements_np = np.zeros(
+                shape=(len(current_element_tags), nodes_per_element),
+                dtype=int
+            )
+            for index, element in zip(element_indices, elements):
+                elements_np[index] = element
+
+            elements_per_type[element_type] = elements_np
+
+        return elements_per_type
 
     def get_mesh_nodes_and_elements(self) -> Tuple[npt.NDArray, npt.NDArray]:
         """Creates the nodes and elements lists as used in the simulation.
@@ -80,7 +148,20 @@ def get_mesh_nodes_and_elements(self) -> Tuple[npt.NDArray, npt.NDArray]:
         Returns:
             List of nodes and elements"""
         nodes = self._get_nodes(*gmsh.model.mesh.getNodes())
-        elements = self._get_elements(*gmsh.model.mesh.getElements())
+
+        el_types, el_tags, el_node_tags = gmsh.model.mesh.getElements(dim=2)
+        elements_per_type = self._get_elements(
+            el_types, el_tags, el_node_tags
+        )
+
+        if len(elements_per_type) != 1:
+            raise ValueError(
+                "The given mesh as more (or less) than one element type for "
+                "dim=2"
+            )
+
+        # There should be only one element type for dim=2
+        elements = elements_per_type[el_types[0]]
 
         return nodes, elements
 
@@ -122,41 +203,42 @@ def get_elements_by_physical_groups(
         self,
         needed_pg_names: List[str]
     ) -> Dict[str, npt.NDArray]:
-        """Returns the elements inside the given physical groups.
+        """Get all triangle elements for the given physical group.
 
         Parameters:
             needed_pg_names: List of names of physical groups for which the
-                triangles are returned.
+                triangle elements shall be returned.
 
         Returns:
-            Dictionary where keys are the pg names and the values are a list
-            of triangles of this physical group."""
-        pg_tags = self.get_nodes_by_physical_groups(needed_pg_names)
-        elements = self._get_elements(*gmsh.model.mesh.getElements())
-        triangle_elements = {}
-        for pg_name, nodes in pg_tags.items():
-            current_triangle_elements = []
-            for check_element in elements:
-                # If at least 2 nodes of the check_element are inside of the
-                # nodes list of the current physical group, then the element
-                # is also part of the physical group.
-                found_count = 0
-
-                if check_element[0] in nodes:
-                    found_count += 1
-                if check_element[1] in nodes:
-                    found_count += 1
-                if check_element[2] in nodes:
-                    found_count += 1
-
-                if found_count > 1:
-                    current_triangle_elements.append(check_element)
-
-            triangle_elements[pg_name] = np.array(
-                current_triangle_elements,
-                dtype=int)
-
-        return triangle_elements
+            dictionary where the key is the physical group name and the value
+            is a list of elements for this pg.
+        """
+        _, elements = self.get_mesh_nodes_and_elements()
+        pg_nodes = self.get_nodes_by_physical_groups(needed_pg_names)
+        element_order = self.element_order
+        nodes_per_line = element_order+1
+        pg_elements = {}
+
+        for pg_name in needed_pg_names:
+            nodes = pg_nodes[pg_name]
+
+            # For every possible element, check if it contains 'enought' nodes
+            # from the physical group -> Then add it to the elements list
+            # TODO Rework this: Check if its also possible to return
+            # element lists containing line elements.
+            current_elements = []
+            for element in elements:
+                count = 0
+                for node_index in element:
+                    if node_index in nodes:
+                        count += 1
+
+                if count >= nodes_per_line:
+                    current_elements.append(element)
+
+            pg_elements[pg_name] = np.array(current_elements)
+
+        return pg_elements
 
     def create_element_post_processing_view(
         self,
diff --git a/plutho/mesh/mesh.py b/plutho/mesh/mesh.py
index c746ae7..6de14a0 100644
--- a/plutho/mesh/mesh.py
+++ b/plutho/mesh/mesh.py
@@ -6,6 +6,7 @@
 
 # Third party libraries
 import gmsh
+import numpy as np
 import numpy.typing as npt
 
 # Local libraries
@@ -22,15 +23,18 @@ class Mesh:
     mesh_file_path: str
     file_version: str
     parser: Union[GmshParser, CustomParser]
+    element_order: int
 
     def __init__(
         self,
         file_path: str,
+        element_order: int
     ):
         if not os.path.isfile(file_path):
             raise IOError(f"Mesh file {file_path} not found.")
 
         self.mesh_file_path = file_path
+        self.element_order = element_order
 
         # Check gmsh version
         # If version2  -> custom gmsh parser
@@ -46,9 +50,9 @@ def __init__(
             self.file_version = version
 
         if version.startswith("2"):
-            self.parser = CustomParser(file_path)
+            self.parser = CustomParser(file_path, element_order)
         else:
-            self.parser = GmshParser(file_path)
+            self.parser = GmshParser(file_path, element_order)
 
     def get_mesh_nodes_and_elements(self) -> Tuple[npt.NDArray, npt.NDArray]:
         return self.parser.get_mesh_nodes_and_elements()
@@ -102,7 +106,8 @@ def generate_rectangular_mesh(
         width: float = 0.005,
         height: float = 0.001,
         mesh_size: float = 0.00015,
-        x_offset: float = 0
+        x_offset: float = 0,
+        element_order: int = 1
     ):
         """Creates a gmsh rectangular mesh given the width, height, the mesh
         size and the x_offset.
@@ -121,6 +126,7 @@ def generate_rectangular_mesh(
             gmsh.initialize()
 
         gmsh.clear()
+        gmsh.option.setNumber("Mesh.ElementOrder", element_order)
 
         corner_points = [
             [x_offset, 0],
diff --git a/plutho/postprocessing.py b/plutho/postprocessing.py
index 4511245..1b74d3e 100644
--- a/plutho/postprocessing.py
+++ b/plutho/postprocessing.py
@@ -7,7 +7,7 @@
 
 # Local libraries
 from plutho.simulation.base import energy_integral_theta, \
-    gradient_local_shape_functions
+    gradient_local_shape_functions_2d
 
 
 def calculate_impedance(
@@ -59,6 +59,7 @@ def calculate_stored_thermal_energy(
     theta: npt.NDArray,
     nodes: npt.NDArray,
     elements: npt.NDArray,
+    element_order: int,
     heat_capacity: float,
     density: float
 ) -> Union[float, npt.NDArray]:
@@ -77,33 +78,33 @@ def calculate_stored_thermal_energy(
     Returns:
         The stored energy either as float or list of floats.
     """
+    points_per_element = int(1/2*(element_order+1)*(element_order+2))
 
     if len(theta.shape) == 2:
         # Need to calculate for every time step
         stored_energies = np.zeros(theta.shape[1])
 
         for time_index in range(theta.shape[1]):
-            for element in elements:
-                dn = gradient_local_shape_functions()
-                node_points = np.array([
-                    [nodes[element[0]][0],
-                     nodes[element[1]][0],
-                     nodes[element[2]][0]],
-                    [nodes[element[0]][1],
-                     nodes[element[1]][1],
-                     nodes[element[2]][1]]
-                ])
-                jacobian = np.dot(node_points, dn.T)
-                jacobian_det = np.linalg.det(jacobian)
-                theta_e = np.array([
-                    theta[element[0], time_index],
-                    theta[element[1], time_index],
-                    theta[element[2], time_index]
-                ])
+            for element_index, element in enumerate(elements):
+                node_points = np.zeros(shape=(2, points_per_element))
+                for node_index in range(points_per_element):
+                    node_points[:, node_index] = [
+                        nodes[element[node_index]][0],
+                        nodes[element[node_index]][1]
+                    ]
+
+                theta_e = np.zeros(points_per_element)
+                for node_index in range(points_per_element):
+                    theta_e[node_index] = theta[
+                        element[node_index],
+                        time_index
+                    ]
+
                 stored_energies[time_index] += energy_integral_theta(
                     node_points,
-                    theta_e
-                ) * 2 * np.pi * jacobian_det * heat_capacity * density
+                    theta_e,
+                    element_order
+                ) * 2 * np.pi * heat_capacity * density
 
         return stored_energies
 
@@ -111,28 +112,27 @@ def calculate_stored_thermal_energy(
         # Only one time step
         stored_energy = 0
 
-        for element in elements:
-            dn = gradient_local_shape_functions()
-            node_points = np.array([
-                [nodes[element[0]][0],
-                 nodes[element[1]][0],
-                 nodes[element[2]][0]],
-                [nodes[element[0]][1],
-                 nodes[element[1]][1],
-                 nodes[element[2]][1]]
-            ])
-            jacobian = np.dot(node_points, dn.T)
-            jacobian_det = np.linalg.det(jacobian)
-            theta_e = np.array([
-                theta[element[0]],
-                theta[element[1]],
-                theta[element[2]]
-            ])
+        for element_index, element in enumerate(elements):
+            node_points = np.zeros(shape=(2, points_per_element))
+            for node_index in range(points_per_element):
+                node_points[:, node_index] = [
+                    nodes[element[node_index]][0],
+                    nodes[element[node_index]][1]
+                ]
+
+            theta_e = np.zeros(points_per_element)
+            for node_index in range(points_per_element):
+                theta_e[node_index] = theta[element[node_index]]
+
             stored_energy += energy_integral_theta(
                 node_points,
-                theta_e
-            ) * 2 * np.pi * jacobian_det * heat_capacity * density
+                theta_e,
+                element_order
+            ) * 2 * np.pi * heat_capacity * density
 
         return stored_energy
 
-    return -1
+    raise ValueError(
+        "Cannot calculate total stored energy for given "
+        f"{len(theta.shape)}-dimensional theta"
+    )
diff --git a/plutho/simulation/base.py b/plutho/simulation/base.py
index 94e10f6..2a350e8 100644
--- a/plutho/simulation/base.py
+++ b/plutho/simulation/base.py
@@ -49,6 +49,7 @@ class MeshData:
     """Contains the mesh data is used in the simulation."""
     nodes: npt.NDArray
     elements: npt.NDArray
+    element_order: int
 
 
 class ExcitationType(Enum):
@@ -74,15 +75,6 @@ def asdict(self):
         return content
 
 
-@dataclass
-class LocalElementData():
-    """Conatins the data of a single element"""
-    node_points: npt.NDArray
-    jacobian: npt.NDArray
-    jacobian_inverted_t: npt.NDArray
-    jacobian_det: float
-
-
 @dataclass
 class MaterialData:
     """Contains the plain material data. Some parameters can either be
@@ -153,91 +145,167 @@ def load_from_file(file_path: str):
 # -------- Local functions and integrals --------
 
 
-def local_shape_functions(*dimensions: float):
-    """Returns the local linear shape functions for the given coordinates for a
-    line or a triangle.
-    The dimension of the returned shape functions depends on the number of
-    coordinates given.
+def local_shape_functions_2d(s, t, element_order):
+    """Returns the local linear shape functions based on a reference triangle
+    with corner points [(0,0), (1,0), (1,1)] for the given coordinates.
 
     Parameters:
-        dimensions: Each parameter corresponds to one coordinate. If only one
-            parameter is given it resembles 2 shape functions along a line. If
-            two coordinates are given it resembles 3 shape functions for a
-            triangle.
+        s: Local s parameter.
+        t: Local t parameter.
+        element_order: Order of the shape functions (1=linear, 2=quadratic,
+            3=cubic).
 
     Returns:
         npt.NDArray: Shape functions depending on the given parameters for a
             line or a triangle
     """
-    dim = len(dimensions)
-    if dim == 1:
-        return np.array([1-dimensions[0], dimensions[0]])
-    elif dim == 2:
-        return np.array([
-            1-dimensions[0]-dimensions[1],
-            dimensions[0],
-            dimensions[1]
-        ])
-    else:
-        raise NotImplementedError(
-            "Function not implemented for dimensions of 3 and above"
-        )
+    L1 = 1-s-t
+    L2 = s
+    L3 = t
+
+    match element_order:
+        case 1:
+            return np.array([L1, L2, L3])
+        case 2:
+            return np.array(
+                [
+                    L1*(2*L1-1),
+                    L2*(2*L2-1),
+                    L3*(2*L3-1),
+                    4*L1*L2,
+                    4*L2*L3,
+                    4*L3*L1
+                ]
+            )
+        case 3:
+            return np.array(
+                [
+                    0.5*L1*(3*L1-1)*(3*L1-2),
+                    0.5*L2*(3*L2-1)*(3*L2-2),
+                    0.5*L3*(3*L3-1)*(3*L3-2),
+                    9/2*L1*L2*(3*L1-1),
+                    9/2*L2*L1*(3*L2-1),
+                    9/2*L2*L3*(3*L2-1),
+                    9/2*L3*L2*(3*L3-1),
+                    9/2*L3*L1*(3*L3-1),
+                    9/2*L1*L3*(3*L1-1),
+                    27*L1*L2*L3
+                ]
+            )
+        case _:
+            raise ValueError(
+                "Shape functions not implemented for element order "
+                f"{element_order}"
+            )
+
 
-def gradient_local_shape_functions(dim: int = 2):
-    """Returns the gradient of the local shape functions. The dimension can be
-    1 for a line or 2 for a triangle.
+def gradient_local_shape_functions_2d(s, t, element_order) -> npt.NDArray:
+    """Returns the gradient of the local shape functions.
+
+    Parameters:
+        s: Local s parameter.
+        t: Local t parameter.
+        element_order: Order of the shape functions (1=linear, 2=quadratic,
+            3=cubic).
 
     Returns:
         npt.NDArray: Gradient of each of the shape functions. Shape: (2,n),
             where n is the number of shape functions. Since currently shape
             functions are always linear: dim=1 -> n=2 and dim=2 -> n=3
     """
-    if dim == 1:
-        print("dim=1")
-        return np.array([
-            [-1, 1],
-            [0, 0]
-        ])
-    elif dim == 2:
-        return np.array([
-            [-1, 1, 0],
-            [-1, 0, 1]
-        ])
-    else:
-        raise NotImplementedError(
-            "Function not implemented for dimensions of 3 and above"
-        )
+    match element_order:
+        case 1:
+            return np.array([
+                [-1, 1, 0],  # d_s
+                [-1, 0, 1]   # d_t
+            ])
+        case 2:
+            return np.array([
+                [ # d_s
+                    -3+4*t+4*s,
+                    4*s-1,
+                    0,
+                    4-8*s-4*t,
+                    4*t,
+                    -4*t
+                ],
+                [ # d_t
+                    -3+4*s+4*t,
+                    0,
+                    4*t-1,
+                    -4*s,
+                    4*s,
+                    4-4*s-8*t
+                ]
+            ])
+        case 3:
+            return np.array([
+                [  # d_s
+                    -13.5*s**2-27*s*t+18*s-13.5*t**2+18*t-5.5,
+                    13.5*s**2-9*s+1,
+                    0,
+                    40.5*s**2+54*s*t-45*s+13.5*t**2-22.5*t+9,
+                    -40.5*s**2-27*s*t+36*s+4.5*t-4.5,
+                    t*(27*s-4.5),
+                    t*(13.5*t-4.5),
+                    t*(4.5-13.5*t),
+                    t*(27*s+27*t-22.5),
+                    27*t*(-2*s-t+1)
+                ],
+                [  # d_t
+                    -13.5*s**2-27*s*t+18*s-13.5*t**2+18*t-5.5,
+                    0,
+                    13.5*t**2-9*t+1,
+                    s*(27*s+27*t-22.5),
+                    s*(4.5-13.5*s),
+                    s*(13.5*s-4.5),
+                    s*(27*t-4.5),
+                    -27*s*t+4.5*s-40.5*t**2+36*t-4.5,
+                    13.5*s**2+54*s*t-22.5*s+40.5*t**2-45*t+9,
+                    27*s*(-s-2*t+1)
+                ]
+            ])
+
+    raise ValueError(
+        "Gradient of shape functions not implemented for element "
+        f"order {element_order}"
+    )
 
 
 def local_to_global_coordinates(
-        node_points: npt.NDArray,
-        *dimensions: float
+    node_points: npt.NDArray,
+    s: float,
+    t: float,
+    element_order: int
 ) -> Any:
     """Transforms the local coordinates given by dimensions using the node
     points to the global coordinates r, z.
+
     Parameters:
         node_points: For a line: [[x1, x2], [y1, y2]] and for a triangle:
             [[x1, x2, x3], [y1, y2, y3]] of
-        dimensions: Local coordinates
+        s: Local s coordinate.
+        t: Local t coordinate.
+        element_order: Order of the shape functions.
 
     Returns:
         npt.NDArray: Global coordinates [r, z]
     """
-    dim = len(dimensions)
-
-    if node_points.shape[1] != dim + 1:
+    if node_points.shape[1] != int(1/2*(element_order+1)*(element_order+2)):
         raise ValueError(
             "The given node point array does not fit the given number of"
             " dimensions"
         )
 
-    return np.dot(node_points, local_shape_functions(*dimensions))
+    return np.dot(node_points, local_shape_functions_2d(s, t, element_order))
 
 
 def b_operator_global(
     node_points: npt.NDArray,
     jacobian_inverted_t: npt.NDArray,
-    *dimensions: float
+    s: float,
+    t: float,
+    element_order: int
 ):
     """Calculates the B operator for the local coordinantes which is needed
     for voigt-notation.
@@ -246,29 +314,30 @@ def b_operator_global(
     Parameters:
         node_points: List of node points [[x1, x2, x3], [y1, y2, y3]] of
             one triangle.
-        s: Local coordinate.
-        t: Local coordinate.
         jacobian_inverted_t: Jacobian matrix inverted and transposed, needed
             for calculation of global derivatives.
+        s: Local coordinate.
+        t: Local coordinate.
+        element_order: Order of the shape functions.
 
     Returns:
         B operator 4x6, for a u aligned like [u1_r, u1_z, u2_r, u2_z, ..].
     """
-    dim = len(dimensions)
+    nodes_per_element = int(1/2*(element_order+1)*(element_order+2))
 
     # Get local shape functions and r (because of theta component)
-    n = local_shape_functions(*dimensions)
-    r = local_to_global_coordinates(node_points, *dimensions)[0]
+    n = local_shape_functions_2d(s, t, element_order)
+    r = local_to_global_coordinates(node_points, s, t, element_order)[0]
 
     # Get gradients of local shape functions (s, t)
-    dn = gradient_local_shape_functions(dim)
+    dn = gradient_local_shape_functions_2d(s, t, element_order)
 
     # Convert to gradients of (r, z) using jacobi matrix
     global_dn = np.dot(jacobian_inverted_t, dn)
 
     # Initialize and fill array
-    b = np.zeros(shape=(4, 2*(dim+1)))
-    for d in range(dim+1):
+    b = np.zeros(shape=(4, 2*nodes_per_element))
+    for d in range(nodes_per_element):
         b[:, 2*d:2*d+2] = [
             [
                 global_dn[0][d], 0
@@ -287,126 +356,150 @@ def b_operator_global(
     return b
 
 
-def integral_m(node_points: npt.NDArray):
+def integral_m(node_points: npt.NDArray, element_order: int):
     """Calculates the M integral.
 
     Parameters:
         node_points: List of node points [[x1, x2, x3], [y1, y2, y3]] of
             one triangle.
+        element_order: Order of the shape functions.
 
     Returns:
         npt.NDArray: 3x3 M matrix for the given element.
     """
     def inner(s, t):
-        n = local_shape_functions(s, t)
+        dn = gradient_local_shape_functions_2d(s, t, element_order)
+        jacobian = np.dot(node_points, dn.T)
+        jacobian_det = np.linalg.det(jacobian)
+
+        n = local_shape_functions_2d(s, t, element_order)
 
         # Since the simulation is axisymmetric it is necessary
         # to multiply with the radius in the integral
         # (for the theta component (azimuth))
-        r = local_to_global_coordinates(node_points, s, t)[0]
+        r = local_to_global_coordinates(node_points, s, t, element_order)[0]
 
         # Get all combinations of shape function multiplied with each other
-        return np.outer(n, n)*r
+        return np.outer(n, n)*r*jacobian_det
 
-    return quadratic_quadrature(inner)
+    return quadratic_quadrature(inner, element_order)
 
 
 def integral_ku(
     node_points: npt.NDArray,
-    jacobian_inverted_t: npt.NDArray,
-    elasticity_matrix: npt.NDArray
+    elasticity_matrix: npt.NDArray,
+    element_order: int
 ):
     """Calculates the Ku integral
 
     Parameters:
         node_points: List of node points [[x1, x2, x3], [y1, y2, y3]] of
             one triangle.
-        jacobian_inverted_t: Jacobian matrix inverted and transposed, needed
-            for calculation of global derivatives.
         elasticity_matrix: Elasticity matrix for the current element
             (c matrix).
+        element_order: Order of the shape functions.
 
     Returns:
         npt.NDArray: 6x6 Ku matrix for the given element.
     """
     def inner(s, t):
+        dn = gradient_local_shape_functions_2d(s, t, element_order)
+        jacobian = np.dot(node_points, dn.T)
+        jacobian_inverted_t = np.linalg.inv(jacobian).T
+        jacobian_det = np.linalg.det(jacobian)
+
         b_op = b_operator_global(
             node_points,
             jacobian_inverted_t,
             s,
-            t
+            t,
+            element_order
         )
-        r = local_to_global_coordinates(node_points, s, t)[0]
+        r = local_to_global_coordinates(node_points, s, t, element_order)[0]
 
-        return np.dot(np.dot(b_op.T, elasticity_matrix), b_op)*r
+        return np.dot(np.dot(b_op.T, elasticity_matrix), b_op)*r*jacobian_det
 
-    return quadratic_quadrature(inner)
+    return quadratic_quadrature(inner, element_order)
 
 
 def integral_kuv(
     node_points: npt.NDArray,
-    jacobian_inverted_t: npt.NDArray,
-    piezo_matrix: npt.NDArray
+    piezo_matrix: npt.NDArray,
+    element_order: int
 ):
     """Calculates the KuV integral.
 
     Parameters:
         node_points: List of node points [[x1, x2, x3], [y1, y2, y3]] of
             one triangle.
-        jacobian_inverted_t: Jacobian matrix inverted and transposed, needed
-            for calculation of global derivatives.
         piezo_matrix: Piezo matrix for the current element (e matrix).
+        element_order: Order of the shape functions.
 
     Returns:
         npt.NDArray: 6x3 KuV matrix for the given element.
     """
     def inner(s, t):
+        dn = gradient_local_shape_functions_2d(s, t, element_order)
+        jacobian = np.dot(node_points, dn.T)
+        jacobian_inverted_t = np.linalg.inv(jacobian).T
+        jacobian_det = np.linalg.det(jacobian)
+
         b_op = b_operator_global(
             node_points,
             jacobian_inverted_t,
             s,
-            t
+            t,
+            element_order
         )
-        dn = gradient_local_shape_functions()
         global_dn = np.dot(jacobian_inverted_t, dn)
-        r = local_to_global_coordinates(node_points, s, t)[0]
+        r = local_to_global_coordinates(node_points, s, t, element_order)[0]
 
-        return np.dot(np.dot(b_op.T, piezo_matrix.T), global_dn)*r
+        return np.dot(np.dot(b_op.T, piezo_matrix.T), global_dn)*r*jacobian_det
 
-    return quadratic_quadrature(inner)
+    return quadratic_quadrature(inner, element_order)
 
 
 def integral_kve(
     node_points: npt.NDArray,
-    jacobian_inverted_t: npt.NDArray,
-    permittivity_matrix: npt.NDArray
+    permittivity_matrix: npt.NDArray,
+    element_order: int
 ):
     """Calculates the KVe integral.
 
     Parameters:
         node_points: List of node points [[x1, x2, x3], [y1, y2, y3]] of
             one triangle.
-        jacobian_inverted_t: Jacobian matrix inverted and transposed, needed
-            for calculation of global derivatives.
         permittivity_matrix: Permittivity matrix for the
             current element (epsilon matrix).
+        element_order: Order of the shape functions.
 
     Returns:
         npt.NDArray: 3x3 KVe matrix for the given element.
     """
     def inner(s, t):
-        dn = gradient_local_shape_functions()
+        dn = gradient_local_shape_functions_2d(s, t, element_order)
+        jacobian = np.dot(node_points, dn.T)
+        jacobian_inverted_t = np.linalg.inv(jacobian).T
+        jacobian_det = np.linalg.det(jacobian)
+
         global_dn = np.dot(jacobian_inverted_t, dn)
-        r = local_to_global_coordinates(node_points, s, t)[0]
+        r = local_to_global_coordinates(node_points, s, t, element_order)[0]
 
-        return np.dot(np.dot(global_dn.T, permittivity_matrix), global_dn)*r
+        return np.dot(
+            np.dot(
+                global_dn.T,
+                permittivity_matrix
+            ),
+            global_dn
+        )*r*jacobian_det
 
-    return quadratic_quadrature(inner)
+    return quadratic_quadrature(inner, element_order)
 
 
 def energy_integral_theta(
     node_points: npt.NDArray,
-    theta: npt.NDArray
+    theta: npt.NDArray,
+    element_order: int
 ):
     """Integrates the given element over the given theta field.
 
@@ -415,17 +508,22 @@ def energy_integral_theta(
             one triangle.
         theta: List of the temperature field values of the points
             [theta1, theta2, theta3].
+        element_order: Order of the shape functions.
     """
     def inner(s, t):
-        n = local_shape_functions(s, t)
-        r = local_to_global_coordinates(node_points, s, t)[0]
+        dn = gradient_local_shape_functions_2d(s, t, element_order)
+        jacobian = np.dot(node_points, dn.T)
+        jacobian_det = np.linalg.det(jacobian)
+
+        n = local_shape_functions_2d(s, t, element_order)
+        r = local_to_global_coordinates(node_points, s, t, element_order)[0]
 
-        return np.dot(n.T, theta)*r
+        return np.dot(n.T, theta)*r*jacobian_det
 
-    return quadratic_quadrature(inner)
+    return quadratic_quadrature(inner, element_order)
 
 
-def integral_volume(node_points: npt.NDArray):
+def integral_volume(node_points: npt.NDArray, element_order: int):
     """Calculates the volume of the triangle given by the node points.
     HINT: Must be multiplied with 2*np.pi and the jacobian determinant in order
     to give the correct volume of any rotationsymmetric triangle.
@@ -433,46 +531,129 @@ def integral_volume(node_points: npt.NDArray):
     Parameters:
         node_points: List of node points [[x1, x2, x3], [y1, y2, y3]] of
             one triangle.
+        element_order: Order of the shape functions.
 
     Returns:
         Float. Volume of the triangle.
     """
     def inner(s, t):
-        r = local_to_global_coordinates(node_points, s, t)[0]
-        return r
+        dn = gradient_local_shape_functions_2d(s, t, element_order)
+        jacobian = np.dot(node_points, dn.T)
+        jacobian_det = np.linalg.det(jacobian)
+
+        r = local_to_global_coordinates(node_points, s, t, element_order)[0]
+
+        return r*jacobian_det
 
-    return quadratic_quadrature(inner)
+    return quadratic_quadrature(inner, element_order)
 
 
-def quadratic_quadrature(func: Callable):
+def quadratic_quadrature(func: Callable, element_order: int):
     """Integrates the given function of 2 variables using gaussian
     quadrature along 2 variables for a reference triangle.
     This gives exact results for linear shape functions.
 
     Parameters:
         func: Function which will be integrated.
+        element_order: Order of the shape functions.
 
     Returns:
         Integral of the given function.
     """
-    weight_1 = 1/6
-    weight_2 = 2/3
-    return func(weight_1, weight_1)*weight_1 + \
-        func(weight_2, weight_1)*weight_1 + \
-        func(weight_1, weight_2)*weight_1
+    weights = []
+    points = []
+
+    match element_order:
+        case 1 | 2:
+            w1 = 1/6
+            w2 = 2/3
+            weights = np.array([w1, w1, w1])
+            points = np.array([
+                [w1, w1],
+                [w2, w1],
+                [w1, w2]
+            ])
+        case 3 | 4 | 5:
+            w1 = 155-np.sqrt(15)
+            w2 = 155+np.sqrt(15)
+            w3 = 2400
+            weights = np.array([
+                w1/w3,
+                w1/w3,
+                w1/w3,
+                w2/w3,
+                w2/w3,
+                w2/w3,
+                9/80
+            ])
+            p1 = 6-np.sqrt(15)
+            p2 = 9+2*np.sqrt(15)
+            p3 = 6+np.sqrt(15)
+            p4 = 9-2*np.sqrt(15)
+            p5 = 21
+            points = np.array([
+                [p1/p5, p1/p5],
+                [p2/p5, p1/p5],
+                [p1/p5, p2/p5],
+                [p3/p5, p3/p5],
+                [p4/p5, p3/p5],
+                [p3/p5, p4/p5],
+                [1/3, 1/3]
+            ])
+        case _:
+            raise NotImplementedError(
+                "No quadratic quadrature for element order "
+                f"{element_order} implemented"
+            )
+
+    # TODO Can this made faster using numpy?
+    sum = 0
+    for i in range(len(weights)):
+        sum += weights[i]*func(points[i][0], points[i][1])
+
+    return sum
 
 
-def line_quadrature(func):
+def line_quadrature(func: Callable, element_order: int):
     """Integrates the given function of 2 variables along one variable
     for a reference triangle.
     This gives exact results for linear shape functions.
 
     Parameters:
-        func: Function which will be integrated along r-axis
+        func: Function which will be integrated along r-axis.
+        element_order: Order of the shape functions.
 
     Returns:
         Integral of the given function along r-axis"""
-    return 0.5*(func(-1/2/np.sqrt(3)+1/2) + func(1/2/np.sqrt(3)+1/2))
+    weights = []
+    points = []
+
+    match element_order:
+        case 1 | 2:
+            weights = np.array([1/2, 1/2])
+            points = np.array([
+                (3-np.sqrt(3))/6,
+                (3+np.sqrt(3))/6
+            ])
+        case 3 | 4 | 5:
+            weights = np.array([5/18, 8/18, 5/18])
+            points = np.array([
+                (5-np.sqrt(15))/10,
+                1/2,
+                (5+np.sqrt(15))/5
+            ])
+        case _:
+            raise NotImplementedError(
+                "No linear quadrature for element order "
+                f"{element_order} implemented"
+            )
+
+    # TODO Make use of numpy?
+    sum = 0
+    for i in range(len(weights)):
+        sum += weights[i]*func(points[i])
+
+    return sum
 
 
 # -------- Boundary condition functions --------
@@ -629,10 +810,11 @@ def create_dirichlet_bc_nodes(
         [dirichlet_values_u, dirichlet_values_v]
 
 
-def create_local_element_data(
+def create_node_points(
     nodes: npt.NDArray,
-    elements: npt.NDArray
-) -> List[LocalElementData]:
+    elements: npt.NDArray,
+    element_order: int
+) -> npt.NDArray:
     """Create the local node data and the corresponding matrices
     for every element which are needed in many parts of the simulations.
 
@@ -643,49 +825,45 @@ def create_local_element_data(
     Returns:
         List of LocalElementData objects.
     """
-    local_element_data = []
-    dn = gradient_local_shape_functions(2)
-    for element in elements:
+    points_per_element = int(1/2*(element_order+1)*(element_order+2))
+
+    node_points = np.zeros(
+        shape=(len(elements), 2, points_per_element)
+    )
+
+    for element_index, element in enumerate(elements):
         # Get node points of element in format
-        # [x1 x2 x3]
-        # [y1 y2 y3] where (xi, yi) are the coordinates for Node i
-        node_points = np.array([
-            [nodes[element[0]][0],
-             nodes[element[1]][0],
-             nodes[element[2]][0]],
-            [nodes[element[0]][1],
-             nodes[element[1]][1],
-             nodes[element[2]][1]]
-        ])
-        jacobian = np.dot(node_points, dn.T)
-        jacobian_inverted_t = np.linalg.inv(jacobian).T
-        jacobian_det = np.linalg.det(jacobian)
-        local_element_data.append(LocalElementData(
-            node_points,
-            jacobian,
-            jacobian_inverted_t,
-            jacobian_det
-        ))
+        # [x1 x2 x3 ... xn]
+        # [y1 y2 y3 ... yn] where (xi, yi) are the coordinates for Node i
+        for node_index in range(points_per_element):
+            node_points[element_index, :, node_index] = [
+                nodes[element[node_index]][0],
+                nodes[element[node_index]][1]
+            ]
 
-    return local_element_data
+    return node_points
 
 
-def calculate_volumes(local_element_data: List[LocalElementData]):
+def calculate_volumes(node_points: npt.NDArray, element_order):
     """Calculates the volume of each element. The element information
     is given by the local_element_data
 
     Parameters:
-        local_element_data: List of LocalElementData objects.
+        node_points: Node points of all elements for which the volume shall be
+            calculated.
 
     Returns:
         List of volumes of the elements.
     """
     volumes = []
 
-    for local_element in local_element_data:
-        node_points = local_element.node_points
-        jacobian_det = local_element.jacobian_det
-        volumes.append(integral_volume(node_points) * 2 * np.pi * jacobian_det)
+    number_of_elements = node_points.shape[0]
+
+    for element_index in range(number_of_elements):
+        volumes.append(
+            integral_volume(node_points[element_index], element_order)
+            *2*np.pi
+        )
 
     return volumes
 
diff --git a/plutho/simulation/nonlinear/base.py b/plutho/simulation/nonlinear/base.py
index e17c8a6..1516793 100644
--- a/plutho/simulation/nonlinear/base.py
+++ b/plutho/simulation/nonlinear/base.py
@@ -12,7 +12,7 @@
 # Local libraries
 from ..base import MeshData, local_to_global_coordinates, b_operator_global, \
     integral_m, integral_ku, integral_kuv, integral_kve, \
-    create_local_element_data, quadratic_quadrature
+    create_node_points, quadratic_quadrature, gradient_local_shape_functions_2d
 from plutho.materials import MaterialManager
 
 
@@ -78,7 +78,8 @@ def assemble(
     # Maybe the 2x2 matrix slicing is not very fast
     nodes = mesh_data.nodes
     elements = mesh_data.elements
-    local_elements = create_local_element_data(nodes, elements)
+    element_order = mesh_data.element_order
+    node_points = create_node_points(nodes, elements, element_order)
 
     number_of_nodes = len(nodes)
     mu = sparse.lil_matrix(
@@ -103,11 +104,7 @@ def assemble(
     )
 
     for element_index, element in enumerate(elements):
-        # Get local element nodes and matrices
-        local_element = local_elements[element_index]
-        node_points = local_element.node_points
-        jacobian_inverted_t = local_element.jacobian_inverted_t
-        jacobian_det = local_element.jacobian_det
+        current_node_points = node_points[element_index]
 
         # TODO Check if its necessary to calculate all integrals
         # --> Dirichlet nodes could be leaved out?
@@ -116,42 +113,39 @@ def assemble(
         # Mutiply with 2*pi because theta is integrated from 0 to 2*pi
         mu_e = (
             material_manager.get_density(element_index)
-            * integral_m(node_points)
-            * jacobian_det * 2 * np.pi
+            * integral_m(current_node_points, element_order)
+            * 2 * np.pi
         )
         ku_e = (
             integral_ku(
-                node_points,
-                jacobian_inverted_t,
-                material_manager.get_elasticity_matrix(element_index)
-            )
-            * jacobian_det * 2 * np.pi
+                current_node_points,
+                material_manager.get_elasticity_matrix(element_index),
+                element_order
+            ) * 2 * np.pi
         )
         kuv_e = (
             integral_kuv(
-                node_points,
-                jacobian_inverted_t,
-                material_manager.get_piezo_matrix(element_index)
-            )
-            * jacobian_det * 2 * np.pi
+                current_node_points,
+                material_manager.get_piezo_matrix(element_index),
+                element_order
+            ) * 2 * np.pi
         )
         kve_e = (
             integral_kve(
-                node_points,
-                jacobian_inverted_t,
+                current_node_points,
                 material_manager.get_permittivity_matrix(
                     element_index
-                )
-            )
-            * jacobian_det * 2 * np.pi
+                ),
+                element_order
+            ) * 2 * np.pi
         )
         if nonlinear_type is NonlinearType.Custom:
             lu_e = (
                 integral_u_nonlinear(
-                    node_points,
-                    jacobian_inverted_t,
-                    nonlinear_matrix
-                ) * jacobian_det * 2 * np.pi
+                    current_node_points,
+                    nonlinear_matrix,
+                    element_order
+                ) * 2 * np.pi
             )
 
         # Now assemble all element matrices
@@ -227,8 +221,8 @@ def assemble(
 
 def integral_u_nonlinear(
     node_points: npt.NDArray,
-    jacobian_inverted_t: npt.NDArray,
-    nonlinear_elasticity_matrix: npt.NDArray
+    nonlinear_elasticity_matrix: npt.NDArray,
+    element_order: int
 ):
     """Calculates the Ku integral
 
@@ -244,13 +238,26 @@ def integral_u_nonlinear(
         npt.NDArray: 6x6 Ku matrix for the given element.
     """
     def inner(s, t):
+        dn = gradient_local_shape_functions_2d(s, t, element_order)
+        jacobian = np.dot(node_points, dn.T)
+        jacobian_inverted_t = np.linalg.inv(jacobian).T
+        jacobian_det = np.linalg.det(jacobian)
+
         b_op = b_operator_global(
-            node_points, jacobian_inverted_t,
+            node_points,
+            jacobian_inverted_t,
             s,
             t,
+            element_order
         )
-        r = local_to_global_coordinates(node_points, s, t)[0]
+        r = local_to_global_coordinates(node_points, s, t, element_order)[0]
 
-        return 1/2*np.dot(np.dot(b_op.T, nonlinear_elasticity_matrix), b_op)*r
+        return np.dot(
+            np.dot(
+                b_op.T,
+                nonlinear_elasticity_matrix
+            ),
+            b_op
+        ) * 1/2 * r * jacobian_det
 
-    return quadratic_quadrature(inner)
+    return quadratic_quadrature(inner, element_order)
diff --git a/plutho/simulation/nonlinear/piezo_stationary.py b/plutho/simulation/nonlinear/piezo_stationary.py
index cc48774..1321636 100644
--- a/plutho/simulation/nonlinear/piezo_stationary.py
+++ b/plutho/simulation/nonlinear/piezo_stationary.py
@@ -3,7 +3,7 @@
 """
 
 # Python standard libraries
-from typing import List, Union, Tuple
+from typing import Union, Tuple
 
 # Third party libraries
 import numpy as np
@@ -14,7 +14,7 @@
 
 # Local libraries
 from .base import assemble, NonlinearType
-from ..base import MeshData, LocalElementData
+from ..base import MeshData
 from plutho.materials import MaterialManager
 
 
@@ -38,9 +38,6 @@ class NonlinearPiezoSimStationary:
     k: sparse.lil_array
     ln: sparse.lil_array
 
-    # Internal simulation data
-    local_elements: List[LocalElementData]
-
     # Resulting fields
     u: npt.NDArray
 
@@ -86,7 +83,7 @@ def solve_linear(self):
 
         # Calculate u using phi as load vector
         self.u = linalg.spsolve(
-            k.tocsc(),
+            k,
             f
         )
 
@@ -244,10 +241,10 @@ def calculate_tangent_matrix_hadamard(
 
     @staticmethod
     def apply_dirichlet_bc(
-        k: sparse.sparray,
-        ln: sparse.sparray,
+        k: sparse.lil_array,
+        ln: sparse.lil_array,
         nodes: npt.NDArray
-    ) -> Tuple[sparse.sparray, sparse.sparray]:
+    ) -> Tuple[sparse.csc_array, sparse.csc_array]:
         """Sets dirichlet boundary condition for the given matrix.
 
         Parameters:
@@ -264,4 +261,4 @@ def apply_dirichlet_bc(
 
             k[node, node] = 1
 
-        return k, ln
+        return k.tocsc(), ln.tocsc()
diff --git a/plutho/simulation/nonlinear/piezo_time.py b/plutho/simulation/nonlinear/piezo_time.py
index 7e118c8..e614b32 100644
--- a/plutho/simulation/nonlinear/piezo_time.py
+++ b/plutho/simulation/nonlinear/piezo_time.py
@@ -1,7 +1,7 @@
 """Module for the simulation of nonlinear piezoelectric systems"""
 
 # Third party libraries
-from typing import List
+from typing import Tuple
 import numpy as np
 import numpy.typing as npt
 import scipy
@@ -10,10 +10,8 @@
 
 # Local libraries
 from .base import assemble, NonlinearType
-from ..base import SimulationData, MeshData, gradient_local_shape_functions, \
-    LocalElementData
-from plutho.simulation.piezo_time import charge_integral_u, \
-    charge_integral_v
+from ..base import SimulationData, MeshData
+from plutho.simulation.piezo_time import calculate_charge
 from plutho.materials import MaterialManager
 
 
@@ -42,7 +40,6 @@ class NonlinearPiezoSimTime:
             mechanical field.
         u: Mechanical field vector u(element, t).
         q: Charge q(t).
-        local_elements: List of element data for the local triangles.
 
     Parameters:
         mesh_data: MeshData object containing the mesh.
@@ -65,9 +62,6 @@ class NonlinearPiezoSimTime:
     u: npt.NDArray
     q: npt.NDArray
 
-    # Internal simulation data
-    local_elements: List[LocalElementData]
-
     # FEM matrices
     m: sparse.lil_array
     c: sparse.lil_array
@@ -264,13 +258,13 @@ def residual(next_u, current_u, v, a, f):
             # Calculate charge
             if (electrode_elements is not None
                     and electrode_elements.shape[0] > 0):
-                q[time_index] = NonlinearPiezoSimTime.calculate_charge(
-                    u[:2*number_of_nodes, time_index+1],
-                    u[2*number_of_nodes:, time_index+1],
+                q[time_index] = calculate_charge(
+                    u[:, time_index+1],
                     self.material_manager,
                     electrode_elements,
                     electrode_normals,
-                    self.mesh_data.nodes
+                    self.mesh_data.nodes,
+                    self.mesh_data.element_order
                 )
 
         self.u = u
@@ -303,86 +297,6 @@ def get_load_vector(
 
         return f
 
-    @staticmethod
-    def calculate_charge(
-        u: npt.NDArray,
-        phi: npt.NDArray,
-        material_manager: MaterialManager,
-        electrode_elements: npt.NDArray,
-        element_normals: npt.NDArray,
-        nodes: npt.NDArray
-    ) -> float:
-        """Calculates the charge using u and phi on the given electrode
-        elements.
-
-        Parameters:
-            u: Mechanical displacement field.
-            phi: Electrical potential field.
-            material_manager: Contains the material data.
-            electrode_elements: Indices of the element on which the charge
-                is calculated.
-            nodes: All nodes from the mesh.
-
-        Returns:
-            The charge on the given elements summed up."""
-        q = 0
-
-        for element_index, element in enumerate(electrode_elements):
-            dn = gradient_local_shape_functions()
-            node_points = np.array([
-                [nodes[element[0]][0],
-                 nodes[element[1]][0],
-                 nodes[element[2]][0]],
-                [nodes[element[0]][1],
-                 nodes[element[1]][1],
-                 nodes[element[2]][1]]
-            ])
-
-            jacobian = np.dot(node_points, dn.T)
-            jacobian_inverted_t = np.linalg.inv(jacobian).T
-            jacobian_det = np.sqrt(
-                np.square(nodes[element[1]][0]-nodes[element[0]][0])
-                + np.square(nodes[element[1]][1]-nodes[element[0]][1])
-            )
-
-            u_e = np.array([
-                u[2*element[0]],
-                u[2*element[0]+1],
-                u[2*element[1]],
-                u[2*element[1]+1],
-                u[2*element[2]],
-                u[2*element[2]+1]
-            ])
-            phi_e = np.array([
-                phi[element[0]],
-                phi[element[1]],
-                phi[element[2]]
-            ])
-
-            q_u = charge_integral_u(
-                node_points,
-                u_e,
-                material_manager.get_piezo_matrix(element_index),
-                jacobian_inverted_t
-            ) * 2 * np.pi * jacobian_det
-            q_v = charge_integral_v(
-                node_points,
-                phi_e,
-                material_manager.get_permittivity_matrix(element_index),
-                jacobian_inverted_t
-            ) * 2 * np.pi * jacobian_det
-
-            # Now take the component normal to the line (outer direction)
-            q_u_normal = np.dot(q_u, element_normals[element_index])
-            q_v_normal = np.dot(q_v, element_normals[element_index])
-
-            q += q_u_normal - q_v_normal
-
-        if np.isnan(q):
-            print("Calculated charge is nan")
-
-        return q
-
     @staticmethod
     def apply_dirichlet_bc(
         m: sparse.lil_array,
@@ -390,7 +304,12 @@ def apply_dirichlet_bc(
         k: sparse.lil_array,
         ln: sparse.lil_array,
         nodes: npt.NDArray
-    ):
+    ) -> Tuple[
+        sparse.csc_array,
+        sparse.csc_array,
+        sparse.csc_array,
+        sparse.csc_array
+    ]:
         # TODO Parameters are not really ndarrays
         # Set rows of matrices to 0 and diagonal of K to 1 (at node points)
 
diff --git a/plutho/simulation/piezo_freq.py b/plutho/simulation/piezo_freq.py
index 96d0c31..24f17f3 100644
--- a/plutho/simulation/piezo_freq.py
+++ b/plutho/simulation/piezo_freq.py
@@ -10,8 +10,8 @@
 # Local libraries
 from .base import MeshData, local_to_global_coordinates, b_operator_global, \
     integral_m, integral_ku, integral_kuv, integral_kve, apply_dirichlet_bc, \
-    create_local_element_data, LocalElementData, quadratic_quadrature, \
-    calculate_volumes
+    create_node_points, quadratic_quadrature, \
+    calculate_volumes, gradient_local_shape_functions_2d
 from .piezo_time import calculate_charge
 from ..materials import MaterialManager
 
@@ -19,8 +19,8 @@
 def loss_integral_scs(
     node_points: npt.NDArray,
     u_e: npt.NDArray,
-    jacobian_inverted_t: npt.NDArray,
-    elasticity_matrix: npt.NDArray
+    elasticity_matrix: npt.NDArray,
+    element_order: int
 ):
     """Calculate sthe integral of dS/dt*c*dS/dt over one triangle in the
     frequency domain for the given frequency.
@@ -31,25 +31,37 @@ def loss_integral_scs(
         u_e: Displacement at this element [u1_r, u1_z, u_2r, u_2z, u_3r, u_3z].
         angular_frequency: Angular frequency at which the loss shall be
             calculated.
-        jacobian_inverted_t: Jacobian matrix inverted and transposed, needed
-            for calculation of global derivates.
-        elasticity_matrix: Elasticity matrix for the current element (c matrix)
+        elasticity_matrix: Elasticity matrix for the current element
+            (c matrix).
+        element_order: Order of the shape functions.
     """
     def inner(s, t):
-        r = local_to_global_coordinates(node_points, s, t)[0]
+        dn = gradient_local_shape_functions_2d(s, t, element_order)
+        jacobian = np.dot(node_points, dn.T)
+        jacobian_inverted_t = np.linalg.inv(jacobian).T
+        jacobian_det = np.linalg.det(jacobian)
+
         b_opt = b_operator_global(
             node_points,
             jacobian_inverted_t,
             s,
-            t
+            t,
+            element_order
         )
+        r = local_to_global_coordinates(node_points, s, t, element_order)[0]
 
         s_e = np.dot(b_opt, u_e)
 
         # return np.dot(dt_s.T, np.dot(elasticity_matrix.T, dt_s))*r
-        return np.dot(np.conjugate(s_e).T, np.dot(elasticity_matrix.T, s_e))*r
+        return np.dot(
+            np.conjugate(s_e).T,
+            np.dot(
+                elasticity_matrix.T,
+                s_e
+            )
+        ) * r * jacobian_det
 
-    return quadratic_quadrature(inner)
+    return quadratic_quadrature(inner, element_order)
 
 
 class PiezoSimFreq:
@@ -99,7 +111,7 @@ class PiezoSimFreq:
     mech_loss: npt.NDArray
 
     # Internal simulation data
-    local_elements: List[LocalElementData]
+    node_points: npt.NDArray
 
     def __init__(
         self,
@@ -120,7 +132,8 @@ def assemble(self):
         # Maybe the 2x2 matrix slicing is not very fast
         nodes = self.mesh_data.nodes
         elements = self.mesh_data.elements
-        self.local_elements = create_local_element_data(nodes, elements)
+        element_order = self.mesh_data.element_order
+        self.node_points = create_node_points(nodes, elements, element_order)
 
         number_of_nodes = len(nodes)
         mu = sparse.lil_matrix(
@@ -141,11 +154,7 @@ def assemble(self):
         )
 
         for element_index, element in enumerate(self.mesh_data.elements):
-            # Get local element nodes and matrices
-            local_element = self.local_elements[element_index]
-            node_points = local_element.node_points
-            jacobian_inverted_t = local_element.jacobian_inverted_t
-            jacobian_det = local_element.jacobian_det
+            node_points = self.node_points[element_index]
 
             # TODO Check if its necessary to calculate all integrals
             # --> Dirichlet nodes could be leaved out?
@@ -154,34 +163,34 @@ def assemble(self):
             # Mutiply with 2*pi because theta is integrated from 0 to 2*pi
             mu_e = (
                 self.material_manager.get_density(element_index)
-                * integral_m(node_points)
-                * jacobian_det * 2 * np.pi
+                * integral_m(node_points, element_order)
+                * 2 * np.pi
             )
             ku_e = (
                 integral_ku(
                     node_points,
-                    jacobian_inverted_t,
-                    self.material_manager.get_elasticity_matrix(element_index)
+                    self.material_manager.get_elasticity_matrix(element_index),
+                    element_order
                 )
-                * jacobian_det * 2 * np.pi
+                * 2 * np.pi
             )
             kuv_e = (
                 integral_kuv(
                     node_points,
-                    jacobian_inverted_t,
-                    self.material_manager.get_piezo_matrix(element_index)
+                    self.material_manager.get_piezo_matrix(element_index),
+                    element_order
                 )
-                * jacobian_det * 2 * np.pi
+                * 2 * np.pi
             )
             kve_e = (
                 integral_kve(
                     node_points,
-                    jacobian_inverted_t,
                     self.material_manager.get_permittivity_matrix(
                         element_index
-                    )
+                    ),
+                    element_order
                 )
-                * jacobian_det * 2 * np.pi
+                * 2 * np.pi
             )
 
             # Now assemble all element matrices
@@ -259,6 +268,9 @@ def solve_frequency(
         frequencies = self.frequencies
         number_of_nodes = len(self.mesh_data.nodes)
         number_of_elements = len(self.mesh_data.elements)
+        element_order = self.mesh_data.element_order
+        points_per_element = int(1/2*(element_order+1)*(element_order+2))
+
         u = np.zeros(
             (3*number_of_nodes, len(frequencies)),
             dtype=np.complex128
@@ -276,7 +288,10 @@ def solve_frequency(
             self.dirichlet_nodes
         )
 
-        volumes = calculate_volumes(self.local_elements)
+        volumes = calculate_volumes(
+            self.node_points,
+            self.mesh_data.element_order
+        )
 
         print(
             f"Starting frequency simulation. There are {len(frequencies)} "
@@ -305,37 +320,32 @@ def solve_frequency(
                     self.material_manager,
                     electrode_elements,
                     electrode_normals,
-                    self.mesh_data.nodes
+                    self.mesh_data.nodes,
+                    self.mesh_data.element_order,
+                    True
                 )
 
             # Calculate mech_loss for every element
             for element_index, element in enumerate(self.mesh_data.elements):
-                # Get local element data
-                local_element = self.local_elements[element_index]
-                node_points = local_element.node_points
-                jacobian_inverted_t = local_element.jacobian_inverted_t
-                jacobian_det = local_element.jacobian_det
+                node_points = self.node_points[element_index]
 
                 # Get field values
-                u_e = np.array([
-                    u[2*element[0], frequency_index],
-                    u[2*element[0]+1, frequency_index],
-                    u[2*element[1], frequency_index],
-                    u[2*element[1]+1, frequency_index],
-                    u[2*element[2], frequency_index],
-                    u[2*element[2]+1, frequency_index]])
+                u_e = np.zeros(2*points_per_element, dtype=np.complex128)
+                for i in range(points_per_element):
+                    u_e[2*i] = u[2*element[i], frequency_index]
+                    u_e[2*i+1] = u[2*element[i]+1, frequency_index]
 
                 if calculate_mech_loss:
                     mech_loss[element_index, frequency_index] = (
                         loss_integral_scs(
                             node_points,
                             u_e,
-                            jacobian_inverted_t,
                             self.material_manager.get_elasticity_matrix(
                                 element_index
-                            )
+                            ),
+                            element_order
                         )
-                        * 2 * np.pi * jacobian_det
+                        * 2 * np.pi
                         * self.material_manager.get_alpha_k(element_index)
                         * 1/volumes[element_index]
                         # TODO Actually this must be multiplied with -1
diff --git a/plutho/simulation/piezo_time.py b/plutho/simulation/piezo_time.py
index 5b8f5fc..b0f834a 100644
--- a/plutho/simulation/piezo_time.py
+++ b/plutho/simulation/piezo_time.py
@@ -9,10 +9,10 @@
 
 # Local libraries
 from .base import SimulationData, MeshData, \
-    gradient_local_shape_functions, \
+    gradient_local_shape_functions_2d, \
     local_to_global_coordinates, b_operator_global, integral_m, \
     integral_ku, integral_kuv, integral_kve, apply_dirichlet_bc, \
-    line_quadrature, create_local_element_data, LocalElementData
+    line_quadrature, create_node_points
 from ..materials import MaterialManager
 
 
@@ -20,7 +20,7 @@ def charge_integral_u(
     node_points: npt.NDArray,
     u_e: npt.NDArray,
     piezo_matrix: npt.NDArray,
-    jacobian_inverted_t: npt.NDArray
+    element_order: int
 ):
     """Calculates the integral of eBu of the given element.
 
@@ -30,31 +30,35 @@ def charge_integral_u(
         u_e: List of u values at the nodes of the triangle
             [u1_r, u1_z, u2_r, u2_z].
         piezo_matrix: Piezo matrix for the current element (e matrix).
-        jacobian_inverted_t: Jacobian matrix inverted and transposed, needed
-            for calculation of global derivatives.
+        element_order: Order of the shape functions.
 
     Returns:
         Float: Integral of eBu of the current triangle.
     """
     def inner(s):
-        r = local_to_global_coordinates(node_points, s, 0)[0]
+        dn = gradient_local_shape_functions_2d(s, 0, element_order)
+        jacobian = np.dot(node_points, dn.T)
+        jacobian_inverted_t = np.linalg.inv(jacobian).T
+
         b_opt_global = b_operator_global(
             node_points,
             jacobian_inverted_t,
             s,
-            0
+            0,
+            element_order
         )
+        r = local_to_global_coordinates(node_points, s, 0, element_order)[0]
 
         return -np.dot(np.dot(piezo_matrix, b_opt_global), u_e)*r
 
-    return line_quadrature(inner)
+    return line_quadrature(inner, element_order)
 
 
 def charge_integral_v(
     node_points: npt.NDArray,
     v_e: npt.NDArray,
     permittivity_matrix: npt.NDArray,
-    jacobian_inverted_t: npt.NDArray
+    element_order: int
 ):
     """Calculates the integral of epsilonBVe of the given element.
 
@@ -65,20 +69,22 @@ def charge_integral_v(
             [v1, v2].
         permittivity_matrix: Permittivity matrix for the current
             element (e matrix).
-        jacobian_inverted_t: Jacobian matrix inverted and transposed, needed
-            for calculation of global derivatives.
+        element_order: Order of the shape functions.
 
     Returns:
         Float: Integral of epsilonBVe of the current triangle.
     """
     def inner(s):
-        r = local_to_global_coordinates(node_points, s, 0)[0]
-        dn = gradient_local_shape_functions()
+        dn = gradient_local_shape_functions_2d(s, 0, element_order)
+        jacobian = np.dot(node_points, dn.T)
+        jacobian_inverted_t = np.linalg.inv(jacobian).T
+
         global_dn = np.dot(jacobian_inverted_t, dn)
+        r = local_to_global_coordinates(node_points, s, 0, element_order)[0]
 
         return -np.dot(np.dot(permittivity_matrix, global_dn), v_e)*r
 
-    return line_quadrature(inner)
+    return line_quadrature(inner, element_order)
 
 
 def calculate_charge(
@@ -86,7 +92,9 @@ def calculate_charge(
     material_manager: MaterialManager,
     elements: npt.NDArray,
     element_normals: npt.NDArray,
-    nodes
+    nodes: npt.NDArray,
+    element_order: int,
+    is_complex: bool = False
 ) -> float:
     """Calculates the charge of the given elements.
 
@@ -97,28 +105,23 @@ def calculate_charge(
         element_normals: List of element normal vectors (index corresponding to
             the elements list).
         nodes: All nodes used in the simulation.
+        element_order: Order of the shape functions.
     """
+    points_per_element = int(1/2*(element_order+1)*(element_order+2))
     number_of_nodes = len(nodes)
     q = 0
 
     for element_index, element in enumerate(elements):
-        dn = gradient_local_shape_functions(2)
-        node_points = np.array([
-            [nodes[element[0]][0],
-             nodes[element[1]][0],
-             nodes[element[2]][0]],
-            [nodes[element[0]][1],
-             nodes[element[1]][1],
-             nodes[element[2]][1]],
-        ])
-        # This calculation of the jacobian does only work for triangle elements
-        # and not for line elements, since for line elements the resulting
-        # jacobian is non invertible.
-        jacobian = np.dot(node_points, dn.T)
-        jacobian_inverted_t = np.linalg.inv(jacobian).T
+        node_points = np.zeros(shape=(2, points_per_element))
+        for node_index in range(points_per_element):
+            node_points[:, node_index] = [
+                nodes[element[node_index]][0],
+                nodes[element[node_index]][1]
+            ]
 
         # The integration is always done along the first two points of the
         # triangle.
+        # TODO Not true for element_order > 1
         jacobian_det = np.sqrt(
             np.square(nodes[element[1]][0]-nodes[element[0]][0])
             + np.square(nodes[element[1]][1]-nodes[element[0]][1])
@@ -129,31 +132,30 @@ def calculate_charge(
         # This is due to the charge calculation needing the derivatives of the
         # shape function (dN_i/dr and dN_i/dz) which are (+/-) 1 along the
         # whole triangle for all 3 points.
-        u_e = np.array([
-            u[2*element[0]],
-            u[2*element[0]+1],
-            u[2*element[1]],
-            u[2*element[1]+1],
-            u[2*element[2]],
-            u[2*element[2]+1]
-        ])
-        ve_e = np.array([
-            u[element[0]+2*number_of_nodes],
-            u[element[1]+2*number_of_nodes],
-            u[element[2]+2*number_of_nodes]
-        ])
+        # TODO Is this true?
+        if is_complex:
+            u_e = np.zeros(2*points_per_element, dtype=np.complex128)
+            ve_e = np.zeros(points_per_element, dtype=np.complex128)
+        else:
+            u_e = np.zeros(2*points_per_element)
+            ve_e = np.zeros(points_per_element)
+
+        for i in range(points_per_element):
+            u_e[2*i] = u[2*element[i]]
+            u_e[2*i+1] = u[2*element[i]+1]
+            ve_e[i] = u[element[i]+2*number_of_nodes]
 
         q_u = charge_integral_u(
             node_points,
             u_e,
             material_manager.get_piezo_matrix(element_index),
-            jacobian_inverted_t
+            element_order
         ) * 2 * np.pi * jacobian_det
         q_v = charge_integral_v(
             node_points,
             ve_e,
             material_manager.get_permittivity_matrix(element_index),
-            jacobian_inverted_t
+            element_order
         ) * 2 * np.pi * jacobian_det
 
         # Now take the component normal to the line (outer direction)
@@ -211,7 +213,7 @@ class PiezoSimTime:
     q: npt.NDArray
 
     # Internal simulation data
-    local_elements: List[LocalElementData]
+    node_points: npt.NDArray
 
     def __init__(
         self,
@@ -232,7 +234,8 @@ def assemble(self):
         # Maybe the 2x2 matrix slicing is not very fast
         nodes = self.mesh_data.nodes
         elements = self.mesh_data.elements
-        self.local_elements = create_local_element_data(nodes, elements)
+        element_order = self.mesh_data.element_order
+        self.node_points = create_node_points(nodes, elements, element_order)
 
         number_of_nodes = len(nodes)
         mu = sparse.lil_matrix(
@@ -253,11 +256,7 @@ def assemble(self):
         )
 
         for element_index, element in enumerate(self.mesh_data.elements):
-            # Get local element nodes and matrices
-            local_element = self.local_elements[element_index]
-            node_points = local_element.node_points
-            jacobian_inverted_t = local_element.jacobian_inverted_t
-            jacobian_det = local_element.jacobian_det
+            node_points = self.node_points[element_index]
 
             # TODO Check if its necessary to calculate all integrals
             # --> Dirichlet nodes could be leaved out?
@@ -266,34 +265,34 @@ def assemble(self):
             # Mutiply with 2*pi because theta is integrated from 0 to 2*pi
             mu_e = (
                 self.material_manager.get_density(element_index)
-                * integral_m(node_points)
-                * jacobian_det * 2 * np.pi
+                * integral_m(node_points, element_order)
+                * 2 * np.pi
             )
             ku_e = (
                 integral_ku(
                     node_points,
-                    jacobian_inverted_t,
-                    self.material_manager.get_elasticity_matrix(element_index)
+                    self.material_manager.get_elasticity_matrix(element_index),
+                    element_order
                 )
-                * jacobian_det * 2 * np.pi
+                * 2 * np.pi
             )
             kuv_e = (
                 integral_kuv(
                     node_points,
-                    jacobian_inverted_t,
-                    self.material_manager.get_piezo_matrix(element_index)
+                    self.material_manager.get_piezo_matrix(element_index),
+                    element_order
                 )
-                * jacobian_det * 2 * np.pi
+                * 2 * np.pi
             )
             kve_e = (
                 integral_kve(
                     node_points,
-                    jacobian_inverted_t,
                     self.material_manager.get_permittivity_matrix(
                         element_index
-                    )
+                    ),
+                    element_order
                 )
-                * jacobian_det * 2 * np.pi
+                * 2 * np.pi
             )
 
             # Now assemble all element matrices
@@ -443,7 +442,8 @@ def solve_time(
                     self.material_manager,
                     electrode_elements,
                     electrode_normals,
-                    self.mesh_data.nodes
+                    self.mesh_data.nodes,
+                    self.mesh_data.element_order
                 )
 
             if (time_index + 1) % 100 == 0:
diff --git a/plutho/simulation/thermo_piezo_time.py b/plutho/simulation/thermo_piezo_time.py
index 438b164..6473271 100644
--- a/plutho/simulation/thermo_piezo_time.py
+++ b/plutho/simulation/thermo_piezo_time.py
@@ -10,11 +10,11 @@
 
 # Local libraries
 from .base import SimulationData, MeshData, \
-    gradient_local_shape_functions, create_local_element_data, \
+    create_node_points, \
     local_to_global_coordinates, b_operator_global, integral_m, \
     integral_ku, integral_kuv, integral_kve, apply_dirichlet_bc, \
-    quadratic_quadrature, LocalElementData, calculate_volumes, \
-    get_avg_temp_field_per_element
+    quadratic_quadrature, calculate_volumes, \
+    gradient_local_shape_functions_2d, get_avg_temp_field_per_element
 from .piezo_time import calculate_charge
 from .thermo_time import integral_ktheta, integral_theta_load
 from ..materials import MaterialManager
@@ -26,8 +26,8 @@ def loss_integral_scs(
     u_e_t_minus_1: npt.NDArray,
     u_e_t_minus_2: npt.NDArray,
     delta_t: float,
-    jacobian_inverted_t: npt.NDArray,
-    elasticity_matrix: npt.NDArray
+    elasticity_matrix: npt.NDArray,
+    element_order: int
 ):
     """Calculates the integral of dS/dt*c*dS/dt over one triangle. Since foward
     difference quotient of second oder is used the last 2 values of e_u are
@@ -47,14 +47,21 @@ def loss_integral_scs(
             for calculation of global derivatives.
         elasticity_matrix: Elasticity matrix for the current element
             (c matrix).
+        element_order: Order of the shape functions.
     """
     def inner(s, t):
-        r = local_to_global_coordinates(node_points, s, t)[0]
+        dn = gradient_local_shape_functions_2d(s, t, element_order)
+        jacobian = np.dot(node_points, dn.T)
+        jacobian_inverted_t = np.linalg.inv(jacobian).T
+        jacobian_det = np.linalg.det(jacobian)
+
+        r = local_to_global_coordinates(node_points, s, t, element_order)[0]
         b_opt = b_operator_global(
             node_points,
             jacobian_inverted_t,
             s,
-            t
+            t,
+            element_order
         )
 
         s_e = np.dot(b_opt, u_e_t)
@@ -62,9 +69,15 @@ def inner(s, t):
         s_e_t_minus_2 = np.dot(b_opt, u_e_t_minus_2)
         dt_s = (3*s_e-4*s_e_t_minus_1+s_e_t_minus_2)/(2*delta_t)
 
-        return np.dot(dt_s.T, np.dot(elasticity_matrix.T, dt_s))*r
+        return np.dot(
+            dt_s.T,
+            np.dot(
+                elasticity_matrix.T,
+                dt_s
+            )
+        ) * r * jacobian_det
 
-    return quadratic_quadrature(inner)
+    return quadratic_quadrature(inner, element_order)
 
 
 class ThermoPiezoSimTime:
@@ -108,9 +121,9 @@ class ThermoPiezoSimTime:
     dirichlet_values: npt.NDArray
 
     # FEM matrices
-    m: npt.NDArray
-    c: npt.NDArray
-    k: npt.NDArray
+    m: sparse.lil_array
+    c: sparse.lil_array
+    k: sparse.lil_array
 
     # Resulting fields
     u: npt.NDArray
@@ -118,7 +131,7 @@ class ThermoPiezoSimTime:
     mech_loss: npt.NDArray
 
     # Internal simulation data
-    local_elements: List[LocalElementData]
+    node_points: npt.NDArray
 
     def __init__(
         self,
@@ -139,7 +152,8 @@ def assemble(self):
         # Maybe the 2x2 matrix slicing is not very fast
         nodes = self.mesh_data.nodes
         elements = self.mesh_data.elements
-        self.local_elements = create_local_element_data(nodes, elements)
+        element_order = self.mesh_data.element_order
+        self.node_points = create_node_points(nodes, elements, element_order)
 
         number_of_nodes = len(nodes)
         mu = sparse.lil_matrix(
@@ -162,11 +176,7 @@ def assemble(self):
             dtype=np.float64)
 
         for element_index, element in enumerate(self.mesh_data.elements):
-            # Get local element nodes and matrices
-            local_element = self.local_elements[element_index]
-            node_points = local_element.node_points
-            jacobian_inverted_t = local_element.jacobian_inverted_t
-            jacobian_det = local_element.jacobian_det
+            node_points = self.node_points[element_index]
 
             # TODO Check if its necessary to calculate all integrals
             # --> Dirichlet nodes could be leaved out?
@@ -175,45 +185,45 @@ def assemble(self):
             # Mutiply with 2*pi because theta is integrated from 0 to 2*pi
             mu_e = (
                 self.material_manager.get_density(element_index)
-                * integral_m(node_points)
-                * jacobian_det * 2 * np.pi
+                * integral_m(node_points, element_order)
+                * 2 * np.pi
             )
             ku_e = (
                 integral_ku(
                     node_points,
-                    jacobian_inverted_t,
-                    self.material_manager.get_elasticity_matrix(element_index))
-                * jacobian_det * 2 * np.pi
+                    self.material_manager.get_elasticity_matrix(element_index),
+                    element_order
+                )
+                * 2 * np.pi
             )
             kuv_e = (
                 integral_kuv(
                     node_points,
-                    jacobian_inverted_t,
-                    self.material_manager.get_piezo_matrix(element_index))
-                * jacobian_det * 2 * np.pi
+                    self.material_manager.get_piezo_matrix(element_index),
+                    element_order
+                )
+                * 2 * np.pi
             )
             kve_e = (
                 integral_kve(
                     node_points,
-                    jacobian_inverted_t,
                     self.material_manager.get_permittivity_matrix(
                         element_index
-                    )
+                    ),
+                    element_order
                 )
-                * jacobian_det * 2 * np.pi
+                * 2 * np.pi
             )
             ctheta_e = (
-                integral_m(node_points)
+                integral_m(node_points, element_order)
                 * self.material_manager.get_density(element_index)
                 * self.material_manager.get_heat_capacity(element_index)
-                * jacobian_det * 2 * np.pi
+                * 2 * np.pi
             )
             ktheta_e = (
-                integral_ktheta(
-                    node_points,
-                    jacobian_inverted_t)
+                integral_ktheta(node_points, element_order)
                 * self.material_manager.get_thermal_conductivity(element_index)
-                * jacobian_det * 2 * np.pi
+                * 2 * np.pi
             )
 
             # Now assemble all element matrices
@@ -303,6 +313,8 @@ def solve_time(
         delta_t = self.simulation_data.delta_t
         elements = self.mesh_data.elements
         nodes = self.mesh_data.nodes
+        element_order = self.mesh_data.element_order
+        points_per_element = int(1/2*(element_order+1)*(element_order+2))
         number_of_elements = len(elements)
         number_of_nodes = len(nodes)
 
@@ -336,7 +348,10 @@ def solve_time(
             dtype=np.float64
         )
 
-        volumes = calculate_volumes(self.local_elements)
+        volumes = calculate_volumes(
+            self.node_points,
+            self.mesh_data.element_order
+        )
 
         if theta_start is not None:
             if len(theta_start) != number_of_nodes:
@@ -410,39 +425,25 @@ def solve_time(
                     self.material_manager,
                     electrode_elements,
                     electrode_normals,
-                    nodes
+                    nodes,
+                    element_order
                 )
 
             # Calculate power_loss
             for element_index, element in enumerate(elements):
-                # Get local element data
-                local_element = self.local_elements[element_index]
-                node_points = local_element.node_points
-                jacobian_inverted_t = local_element.jacobian_inverted_t
-                jacobian_det = local_element.jacobian_det
-
-                # Get field values at current element at time index
-                u_e = np.array([
-                    u[2*element[0], time_index+1],
-                    u[2*element[0]+1, time_index+1],
-                    u[2*element[1], time_index+1],
-                    u[2*element[1]+1, time_index+1],
-                    u[2*element[2], time_index+1],
-                    u[2*element[2]+1, time_index+1]])
-                u_e_t_minus_1 = np.array([
-                    u[2*element[0], time_index],
-                    u[2*element[0]+1, time_index],
-                    u[2*element[1], time_index],
-                    u[2*element[1]+1, time_index],
-                    u[2*element[2], time_index],
-                    u[2*element[2]+1, time_index]])
-                u_e_t_minus_2 = np.array([
-                    u[2*element[0], time_index-1],
-                    u[2*element[0]+1, time_index-1],
-                    u[2*element[1], time_index-1],
-                    u[2*element[1]+1, time_index-1],
-                    u[2*element[2], time_index-1],
-                    u[2*element[2]+1, time_index-1]])
+                node_points = self.node_points[element_index]
+
+                # Get field values at current element at specific time index
+                u_e = np.zeros(2*points_per_element)
+                u_e_t_minus_1 = np.zeros(2*points_per_element)
+                u_e_t_minus_2 = np.zeros(2*points_per_element)
+                for i in range(points_per_element):
+                    u_e[2*i] = u[2*element[i], time_index+1]
+                    u_e[2*i+1] = u[2*element[i]+1, time_index+1]
+                    u_e_t_minus_1[2*i] = u[2*element[i], time_index]
+                    u_e_t_minus_1[2*i+1] = u[2*element[i]+1, time_index]
+                    u_e_t_minus_2[2*i] = u[2*element[i], time_index-1]
+                    u_e_t_minus_2[2*i+1] = u[2*element[i]+1, time_index-1]
 
                 # The mech loss of the element is divided by the volume
                 # because it must be a power density.
@@ -454,12 +455,12 @@ def solve_time(
                             u_e_t_minus_1,
                             u_e_t_minus_2,
                             delta_t,
-                            jacobian_inverted_t,
                             self.material_manager.get_elasticity_matrix(
                                 element_index
-                            )
+                            ),
+                            element_order
                         )
-                        * 2 * np.pi * jacobian_det
+                        * 2 * np.pi
                         * self.material_manager.get_alpha_k(element_index)
                         * 1/volumes[element_index]
                     )
@@ -496,6 +497,8 @@ def get_load_vector(
         # For the temperature field the load vector represents the temperature
         # sources given by the mechanical losses.
         nodes = self.mesh_data.nodes
+        element_order = self.mesh_data.element_order
+        points_per_element = int(1/2*(element_order+1)*(element_order+2))
         number_of_nodes = len(nodes)
 
         # Can be initialized to 0 because external load and volume charge
@@ -512,21 +515,18 @@ def get_load_vector(
         f_theta = np.zeros(number_of_nodes)
 
         for element_index, element in enumerate(self.mesh_data.elements):
-            dn = gradient_local_shape_functions()
-            node_points = np.array([
-                [nodes[element[0]][0],
-                 nodes[element[1]][0],
-                 nodes[element[2]][0]],
-                [nodes[element[0]][1],
-                 nodes[element[1]][1],
-                 nodes[element[2]][1]]
-            ])
-            jacobian = np.dot(node_points, dn.T)
-            jacobian_det = np.linalg.det(jacobian)
+            node_points = np.zeros(shape=(2, points_per_element))
+            for node_index in range(points_per_element):
+                node_points[:, node_index] = [
+                    nodes[element[node_index]][0],
+                    nodes[element[node_index]][1]
+                ]
 
             f_theta_e = integral_theta_load(
                 node_points,
-                mech_loss_density[element_index])*2*np.pi*jacobian_det
+                mech_loss_density[element_index],
+                element_order
+            ) * 2 * np.pi
 
             for local_p, global_p in enumerate(element):
                 f_theta[global_p] += f_theta_e[local_p]
diff --git a/plutho/simulation/thermo_time.py b/plutho/simulation/thermo_time.py
index 4611517..67343b2 100644
--- a/plutho/simulation/thermo_time.py
+++ b/plutho/simulation/thermo_time.py
@@ -11,77 +11,92 @@
 
 # Local libraries
 from .base import SimulationData, MeshData, \
-    local_shape_functions, gradient_local_shape_functions, \
-    local_to_global_coordinates, integral_m, LocalElementData, \
-    quadratic_quadrature, line_quadrature, create_local_element_data, \
+    local_shape_functions_2d, gradient_local_shape_functions_2d, \
+    local_to_global_coordinates, integral_m, \
+    quadratic_quadrature, line_quadrature, create_node_points, \
     apply_dirichlet_bc, get_avg_temp_field_per_element
 from ..materials import MaterialManager
 
 
 def integral_heat_flux(
     node_points: npt.NDArray,
-    heat_flux: npt.NDArray
+    heat_flux: npt.NDArray,
+    element_order: int
 ):
     """Integrates the heat flux using the shape functions.
 
     Parameters:
-        node_points: List of node points [[x1, x2, ..], [y1, y2, ..]]
-        heat_flux: Heat flux at the points
+        node_points: List of node points [[x1, x2, ..], [y1, y2, ..]].
+        heat_flux: Heat flux at the points.
+        element_order: Order of the shape functions.
 
     Returns:
-        npt.NDArray heat flux integral on each point."""
+        npt.NDArray heat flux integral on each point.
+    """
     def inner(s):
-        n = local_shape_functions(s)
-        r = local_to_global_coordinates(node_points, s)[0]
+        n = local_shape_functions_2d(s, 0, element_order)
+        r = local_to_global_coordinates(node_points, s, 0, element_order)[0]
 
         return n*heat_flux*r
 
-    return line_quadrature(inner)
+    return line_quadrature(inner, element_order)
 
 
 def integral_ktheta(
     node_points: npt.NDArray,
-    jacobian_inverted_t: npt.NDArray
+    element_order: int
 ):
     """Calculates the Ktheta integral.
 
     Parameters:
-        node_points: List of node points [[x1, x2, ..], [y1, y2, ..]]
-        jacobian_inverted_t: Jacobian matrix inverted and transposed, needed
-            for calculation of global derivatives
+        node_points: List of node points [[x1, x2, ..], [y1, y2, ..]].
+        element_order: Order of the shape functions.
 
     Returns:
         npt.NDArray: 3x3 Ktheta matrix for the given element.
     """
     def inner(s, t):
-        dn = gradient_local_shape_functions()
+        dn = gradient_local_shape_functions_2d(s, t, element_order)
+        jacobian = np.dot(node_points, dn.T)
+        jacobian_inverted_t = np.linalg.inv(jacobian).T
+        jacobian_det = np.linalg.det(jacobian)
+
         global_dn = np.dot(jacobian_inverted_t, dn)
-        r = local_to_global_coordinates(node_points, s, t)[0]
+        r = local_to_global_coordinates(node_points, s, t, element_order)[0]
 
-        return np.dot(global_dn.T, global_dn)*r
+        return np.dot(global_dn.T, global_dn)*r*jacobian_det
 
-    return quadratic_quadrature(inner)
+    return quadratic_quadrature(inner, element_order)
 
 
 def integral_theta_load(
-        node_points: npt.NDArray,
-        mech_loss: float):
+    node_points: npt.NDArray,
+    mech_loss: float,
+    element_order: int
+):
     """Returns the load value for the temperature field (f) for the specific
     element.
 
     Parameters:
         node_points: List of node points [[x1, x2, x3], [y1, y2, y3]] of
             one triangle.
-        point_loss: Loss power on each node (heat source)
+        point_loss: Loss power on each node (heat source).
+        element_order: Order of the shape functions.
+
     Returns:
         npt.NDArray: f vector value at the specific ndoe
     """
     def inner(s, t):
-        n = local_shape_functions(s, t)
-        r = local_to_global_coordinates(node_points, s, t)[0]
-        return n*mech_loss*r
+        dn = gradient_local_shape_functions_2d(s, t, element_order)
+        jacobian = np.dot(node_points, dn.T)
+        jacobian_det = np.linalg.det(jacobian)
 
-    return quadratic_quadrature(inner)
+        n = local_shape_functions_2d(s, t, element_order)
+        r = local_to_global_coordinates(node_points, s, t, element_order)[0]
+
+        return n*mech_loss*r*jacobian_det
+
+    return quadratic_quadrature(inner, element_order)
 
 
 class ThermoSimTime:
@@ -105,6 +120,10 @@ class ThermoSimTime:
     theta: npt.NDArray
     f: npt.NDArray
 
+    # FEM Matrices
+    k: sparse.lil_array
+    c: sparse.lil_array
+
     # Convective bc
     enable_convection_bc: bool
     convective_b_e: List[int]
@@ -112,7 +131,7 @@ class ThermoSimTime:
     convective_outer_temp: float
 
     # Internal simulation data
-    local_elements: List[LocalElementData]
+    node_points: npt.NDArray
 
     # Dirichlet data
     dirichlet_nodes: npt.NDArray
@@ -143,7 +162,8 @@ def assemble(self):
         # Maybe the 2x2 matrix slicing is not very fast
         nodes = self.mesh_data.nodes
         elements = self.mesh_data.elements
-        self.local_elements = create_local_element_data(nodes, elements)
+        element_order = self.mesh_data.element_order
+        self.node_points = create_node_points(nodes, elements, element_order)
 
         number_of_nodes = len(nodes)
         c = sparse.lil_matrix(
@@ -153,35 +173,22 @@ def assemble(self):
             (number_of_nodes, number_of_nodes),
             dtype=np.float64)
 
-        for element in self.mesh_data.elements:
-            dn = gradient_local_shape_functions()
-            # Get node points of element in format
-            # [x1 x2 x3]
-            # [y1 y2 y3] where (xi, yi) are the coordinates for Node i
-            node_points = np.array([
-                [nodes[element[0]][0],
-                 nodes[element[1]][0],
-                 nodes[element[2]][0]],
-                [nodes[element[0]][1],
-                 nodes[element[1]][1],
-                 nodes[element[2]][1]]
-            ])
-            jacobian = np.dot(node_points, dn.T)
-            jacobian_inverted_t = np.linalg.inv(jacobian).T
-            jacobian_det = np.linalg.det(jacobian)
+        for element_index, element in enumerate(self.mesh_data.elements):
+            node_points = self.node_points[element_index]
 
             ctheta_e = (
-                integral_m(node_points)
-                * self.material_manager.get_density(element)
-                * self.material_manager.get_heat_capacity(element)
-                * jacobian_det * 2 * np.pi
+                integral_m(node_points, element_order)
+                * self.material_manager.get_density(element_index)
+                * self.material_manager.get_heat_capacity(element_index)
+                * 2 * np.pi
             )
             ktheta_e = (
                 integral_ktheta(
                     node_points,
-                    jacobian_inverted_t)
-                * self.material_manager.get_thermal_conductivity(element)
-                * jacobian_det * 2 * np.pi
+                    element_order
+                )
+                * self.material_manager.get_thermal_conductivity(element_index)
+                * 2 * np.pi
             )
 
             # Now assemble all element matrices
@@ -193,47 +200,6 @@ def assemble(self):
         self.c = c.tolil()
         self.k = k.tolil()
 
-    def set_volume_heat_source(
-        self,
-        mech_loss_density,
-        number_of_time_steps
-    ):
-        """Sets the excitation for the heat conduction simulation.
-        The mech_losses are set for every time step.
-
-        Parameters:
-            mech_losses: The mechanical losses for each element for every
-                time step.
-            number_of_time_steps: Total number of time steps.
-        """
-        nodes = self.mesh_data.nodes
-        number_of_nodes = len(nodes)
-        f = np.zeros(shape=(number_of_nodes, number_of_time_steps))
-        for time_step in range(number_of_time_steps):
-            for element_index, element in enumerate(self.mesh_data.elements):
-                dn = gradient_local_shape_functions()
-                node_points = np.array([
-                    [nodes[element[0]][0],
-                     nodes[element[1]][0],
-                     nodes[element[2]][0]],
-                    [nodes[element[0]][1],
-                     nodes[element[1]][1],
-                     nodes[element[2]][1]]
-                ])
-                jacobian = np.dot(node_points, dn.T)
-                jacobian_det = np.linalg.det(jacobian)
-
-                # TODO Maybe the outer time step loop can be removed
-                f_theta_e = integral_theta_load(
-                    node_points,
-                    mech_loss_density[element_index, time_step]
-                ) * 2 * np.pi * jacobian_det
-
-                for local_p, global_p in enumerate(element):
-                    f[global_p, time_step] += f_theta_e[local_p]
-
-        self.f += f
-
     def set_constant_volume_heat_source(
         self,
         mech_loss_density: npt.NDArray,
@@ -245,28 +211,27 @@ def set_constant_volume_heat_source(
         Parameters:
             mech_loss_density: The mechanical losses for each element. They
                 will be applied for every time step.
-            number_of_time_steps: Total number of time steps
+            number_of_time_steps: Total number of time steps.
         """
         nodes = self.mesh_data.nodes
+        element_order = self.mesh_data.element_order
         number_of_nodes = len(nodes)
-        f = np.zeros(number_of_nodes)
+        points_per_element = int(1/2*(element_order+1)*(element_order+2))
 
+        f = np.zeros(number_of_nodes)
         for element_index, element in enumerate(self.mesh_data.elements):
-            dn = gradient_local_shape_functions()
-            node_points = np.array([
-                [nodes[element[0]][0],
-                 nodes[element[1]][0],
-                 nodes[element[2]][0]],
-                [nodes[element[0]][1],
-                 nodes[element[1]][1],
-                 nodes[element[2]][1]]
-            ])
-            jacobian = np.dot(node_points, dn.T)
-            jacobian_det = np.linalg.det(jacobian)
+            node_points = np.zeros(shape=(2, points_per_element))
+            for node_index in range(points_per_element):
+                node_points[:, node_index] = [
+                    nodes[element[node_index]][0],
+                    nodes[element[node_index]][1]
+                ]
 
             f_theta_e = integral_theta_load(
                 node_points,
-                mech_loss_density[element_index])*2*np.pi*jacobian_det
+                mech_loss_density[element_index],
+                element_order
+            ) * 2 * np.pi
 
             for local_p, global_p in enumerate(element):
                 f[global_p] += f_theta_e[local_p]
@@ -280,7 +245,8 @@ def _calculate_convection_bc(
         boundary_elements: npt.NDArray,
         theta: npt.NDArray,
         alpha: float,
-        outer_temperature: float
+        outer_temperature: float,
+        element_order: int
     ) -> npt.NDArray:
         """Calculates the load vector for the convection boundary condition.
         The condition is calculated using the temeprature at the given
@@ -296,27 +262,46 @@ def _calculate_convection_bc(
             alpha: Heat transfer coefficient.
             outer_temperature: Temperature outside of the model.
         """
-        f = np.zeros(len(nodes))
+        points_per_element = int(1/2*(element_order+1)*(element_order+2))
 
-        for _, element in enumerate(boundary_elements):
-            node_points = np.array([
-                [nodes[element[0]][0],
-                 nodes[element[1]][0]],
-                [nodes[element[0]][1],
-                 nodes[element[1]][1]]
-            ])
+        f = np.zeros(len(nodes))
+        for element_index, element in enumerate(boundary_elements):
+            node_points = np.zeros(shape=(2, points_per_element))
+            for node_index in range(points_per_element):
+                node_points[element_index, :, node_index] = [
+                    nodes[element[node_index]][0],
+                    nodes[element[node_index]][1]
+                ]
+            # TODO Only works for element_order > 1
             jacobian_det = np.sqrt(
                 np.square(nodes[element[1]][0]-nodes[element[0]][0])
                 + np.square(nodes[element[1]][1]-nodes[element[0]][1])
             )
-            theta_e = np.array([
-                theta[element[0]],
-                theta[element[1]]
-            ])
 
-            heat_flux = alpha*(theta_e-np.ones(2)*outer_temperature)
+            # Temperature at boundary
+            boundary_nodes = []
+            match element_order:
+                case 1:
+                    boundary_nodes = np.array([0, 1])
+                case 2:
+                    boundary_nodes = np.array([0, 3, 1])
+                case 3:
+                    boundary_nodes = np.array([0, 3, 4, 1])
+                case _:
+                    raise NotImplementedError(
+                        "Convection boundary condition"
+                        " not implemented for element order > 3"
+                    )
+
+            theta_e = np.zeros(boundary_nodes.shape[0])
+            for index, bnode in enumerate(boundary_nodes):
+                theta_e[index] = element[bnode]
+
+            heat_flux = alpha*(
+                theta_e - np.ones(boundary_nodes.shape[0])*outer_temperature
+            )
             f_e = (
-                integral_heat_flux(node_points, heat_flux)
+                integral_heat_flux(node_points, heat_flux, element_order)
                 * 2 * np.pi * jacobian_det
             )
 
@@ -426,7 +411,8 @@ def solve_time(
                     self.convective_b_e,
                     theta[:, time_index],
                     self.convective_alpha,
-                    self.convective_outer_temp
+                    self.convective_outer_temp,
+                    self.mesh_data.element_order
                 )
 
             # Perform Newmark method
@@ -457,7 +443,7 @@ def solve_until_material_parameters_change(
         self,
         initial_theta_field: npt.NDArray,
         initial_time_step: int
-    ):
+    ) -> int:
         """Runs the simulation using the assembled c and k matrices as well
         as the set excitation.
         Calculates the temperature field and saves it in theta.
@@ -510,12 +496,13 @@ def solve_until_material_parameters_change(
             # Add a convection boundary condition if it set
             if self.enable_convection_bc:
                 f += self._calculate_convection_bc(
-                    self.mesh_data.nodes,
-                    self.convective_b_e,
-                    self.theta[:, time_index],
-                    self.convective_alpha,
-                    self.convective_outer_temp
-                )
+                self.mesh_data.nodes,
+                self.convective_b_e,
+                self.theta[:, time_index],
+                self.convective_alpha,
+                self.convective_outer_temp,
+                self.mesh_data.element_order
+            )
 
             # Perform Newmark method
             # Predictor step
@@ -547,4 +534,5 @@ def solve_until_material_parameters_change(
             if self.material_manager.update_temperature(
                     temp_field_per_element):
                 return time_index+1
+
         return number_of_time_steps
diff --git a/plutho/single_sim.py b/plutho/single_sim.py
index a69204a..3bbbdf6 100644
--- a/plutho/single_sim.py
+++ b/plutho/single_sim.py
@@ -86,7 +86,7 @@ def __init__(
         self.mesh = mesh
 
         nodes, elements = mesh.get_mesh_nodes_and_elements()
-        self.mesh_data = MeshData(nodes, elements)
+        self.mesh_data = MeshData(nodes, elements, mesh.element_order)
         self.material_manager = MaterialManager(len(elements))
         self.charge_calculated = False
         self.mech_loss_calculated = False
diff --git a/pyproject.toml b/pyproject.toml
index 2a233d8..0a4ec32 100644
--- a/pyproject.toml
+++ b/pyproject.toml
@@ -19,7 +19,7 @@ dependencies = [
     "numpy",
     "matplotlib",
     "scipy>=1.12.0",
-    "gmsh",
+    "gmsh==4.13.1",
     "python-dotenv",
     "pyyaml",
     "flake8>=7.2.0",
@@ -51,4 +51,5 @@ plutho = { workspace = true }
 [dependency-groups]
 dev = [
     "plutho",
+    "pytest>=8.4.1",
 ]
diff --git a/tests/data/piezo_freq/q.npy b/tests/data/piezo_freq/q.npy
index 3bf3155..a40a022 100644
Binary files a/tests/data/piezo_freq/q.npy and b/tests/data/piezo_freq/q.npy differ
diff --git a/tests/data/piezo_freq/u.npy b/tests/data/piezo_freq/u.npy
index f8389f1..e8751f2 100644
Binary files a/tests/data/piezo_freq/u.npy and b/tests/data/piezo_freq/u.npy differ
diff --git a/tests/data/piezo_time/q.npy b/tests/data/piezo_time/q.npy
index a0a585a..805b25b 100644
Binary files a/tests/data/piezo_time/q.npy and b/tests/data/piezo_time/q.npy differ
diff --git a/tests/data/piezo_time/u.npy b/tests/data/piezo_time/u.npy
index 67083bc..0be3d4b 100644
Binary files a/tests/data/piezo_time/u.npy and b/tests/data/piezo_time/u.npy differ
diff --git a/tests/data/thermo_piezo_time/mech_loss.npy b/tests/data/thermo_piezo_time/mech_loss.npy
index f73ba23..272087a 100644
Binary files a/tests/data/thermo_piezo_time/mech_loss.npy and b/tests/data/thermo_piezo_time/mech_loss.npy differ
diff --git a/tests/data/thermo_piezo_time/q.npy b/tests/data/thermo_piezo_time/q.npy
index 5e5dbdd..6dd67be 100644
Binary files a/tests/data/thermo_piezo_time/q.npy and b/tests/data/thermo_piezo_time/q.npy differ
diff --git a/tests/data/thermo_piezo_time/u.npy b/tests/data/thermo_piezo_time/u.npy
index 6aa166a..59cbec9 100644
Binary files a/tests/data/thermo_piezo_time/u.npy and b/tests/data/thermo_piezo_time/u.npy differ
diff --git a/tests/test_simulations.py b/tests/test_fields.py
similarity index 81%
rename from tests/test_simulations.py
rename to tests/test_fields.py
index 424511f..f5b00f1 100644
--- a/tests/test_simulations.py
+++ b/tests/test_fields.py
@@ -10,8 +10,8 @@
 # -------- Global variables --------
 
 NUMBER_OF_TIME_STEPS = 1000
-MAX_ERROR = 1e-15
-
+ATOL = 1e-15
+RTOL = 1e-7
 
 pic255 = plutho.MaterialData(
     **{
@@ -34,15 +34,6 @@
     }
 )
 
-# -------- Helper functions --------
-
-
-def compare_numpy_array(a, b):
-    if a.shape != b.shape:
-        return False
-
-    return np.allclose(a, b, atol=MAX_ERROR)
-
 
 # -------- Test functions --------
 
@@ -52,9 +43,13 @@ def test_thermo_time(tmp_path):
     the simulation and compares it with the input energy.
     """
     # Create and load mesh; TODO maybe use smaller mesh size?
+    element_order = 1
     mesh_path = os.path.join(tmp_path, "default_mesh.msh")
-    plutho.Mesh.generate_rectangular_mesh(mesh_path)
-    mesh = plutho.Mesh(mesh_path)
+    plutho.Mesh.generate_rectangular_mesh(
+        mesh_path,
+        element_order=element_order
+    )
+    mesh = plutho.Mesh(mesh_path, element_order)
 
     sim = plutho.SingleSimulation(
         tmp_path,
@@ -68,6 +63,9 @@ def test_thermo_time(tmp_path):
     nodes, elements = mesh.get_mesh_nodes_and_elements()
     number_of_elements = len(elements)
 
+    print(f"Nodes {nodes}")
+    print(f"Elements {elements}")
+
     sim.setup_thermo_time_domain(
         delta_t,
         NUMBER_OF_TIME_STEPS,
@@ -93,12 +91,16 @@ def test_thermo_time(tmp_path):
         sim.solver.theta[:, -1],
         nodes,
         elements,
+        element_order,
         sim.material_manager.get_heat_capacity(0),
         sim.material_manager.get_density(0)
     )
 
     volume = np.sum(
-        plutho.simulation.base.calculate_volumes(sim.solver.local_elements)
+        plutho.simulation.base.calculate_volumes(
+            sim.solver.node_points,
+            element_order
+        )
     )
 
     input_energy = INPUT_POWER_DENSITY*volume
@@ -113,9 +115,13 @@ def test_piezo_time(tmp_path, test=True):
     """Test function for the piezo time domain simulation.
     Tests the simulation for a triangular excitation."""
     # Create and load mesh; TODO maybe use smaller mesh size?
+    element_order = 1
     mesh_path = os.path.join(tmp_path, "default_mesh.msh")
-    plutho.Mesh.generate_rectangular_mesh(mesh_path)
-    mesh = plutho.Mesh(mesh_path)
+    plutho.Mesh.generate_rectangular_mesh(
+        mesh_path,
+        element_order=element_order
+    )
+    mesh = plutho.Mesh(mesh_path, element_order)
 
     sim = plutho.SingleSimulation(
         tmp_path,
@@ -186,19 +192,30 @@ def test_piezo_time(tmp_path, test=True):
 
         # Compare arrays
         # For displacement just take last time step. TODO Is this sufficient?
-        assert compare_numpy_array(uut_u, test_u), \
-            "Displacement u is not equal"
-        assert compare_numpy_array(uut_q, test_q), "Charge is not equal"
+        np.testing.assert_allclose(
+            uut_u,
+            test_u,
+            rtol=RTOL,
+            atol=ATOL,
+            err_msg="Displacement u is not equal"
+        )
+        np.testing.assert_allclose(
+            uut_q,
+            test_q,
+            rtol=RTOL,
+            atol=ATOL,
+            err_msg="Charge is not equal"
+        )
 
     else:
         # Save results
         np.save(
             os.path.join(tmp_path, test_folder_name, "q.npy"),
-            sim.solver.q
+            uut_q
         )
         np.save(
             os.path.join(tmp_path, test_folder_name, "u.npy"),
-            sim.solver.u[:, -1]
+            uut_u
         )
 
 
@@ -207,9 +224,13 @@ def test_piezo_freq(tmp_path, test=True):
     displacement field and charge for a sinusoidal signal.
     """
     # Create and load mesh; TODO maybe use smaller mesh size?
+    element_order = 1
     mesh_path = os.path.join(tmp_path, "default_mesh.msh")
-    plutho.Mesh.generate_rectangular_mesh(mesh_path)
-    mesh = plutho.Mesh(mesh_path)
+    plutho.Mesh.generate_rectangular_mesh(
+        mesh_path,
+        element_order=element_order
+    )
+    mesh = plutho.Mesh(mesh_path, element_order)
 
     sim = plutho.SingleSimulation(
         tmp_path,
@@ -271,19 +292,30 @@ def test_piezo_freq(tmp_path, test=True):
 
         # Compare arrays
         # For displacement just take last time step. TODO Is this sufficient?
-        assert compare_numpy_array(uut_u, test_u), \
-            "Displacement u is not equal"
-        assert compare_numpy_array(uut_q, test_q), "Charge is not equal"
+        np.testing.assert_allclose(
+            uut_u,
+            test_u,
+            rtol=RTOL,
+            atol=ATOL,
+            err_msg="Displacement u is not equal"
+        )
+        np.testing.assert_allclose(
+            uut_q,
+            test_q,
+            rtol=RTOL,
+            atol=ATOL,
+            err_msg="Charge is not equal"
+        )
 
     else:
         # Save test data
         np.save(
             os.path.join(tmp_path, test_folder_name, "q.npy"),
-            sim.solver.q
+            uut_q
         )
         np.save(
             os.path.join(tmp_path, test_folder_name, "u.npy"),
-            sim.solver.u[:, -1]
+            uut_u
         )
 
 
@@ -294,9 +326,13 @@ def test_thermo_piezo_time(tmp_path, test=True):
     last time step are compared with fixed results.
     """
     # Create and load mesh; TODO maybe use smaller mesh size?
+    element_order = 1
     mesh_path = os.path.join(tmp_path, "default_mesh.msh")
-    plutho.Mesh.generate_rectangular_mesh(mesh_path)
-    mesh = plutho.Mesh(mesh_path)
+    plutho.Mesh.generate_rectangular_mesh(
+        mesh_path,
+        element_order=element_order
+    )
+    mesh = plutho.Mesh(mesh_path, element_order)
 
     sim = plutho.SingleSimulation(
         tmp_path,
@@ -371,7 +407,7 @@ def test_thermo_piezo_time(tmp_path, test=True):
 
     # Calculate total loss energy
     volumes = plutho.simulation.base.calculate_volumes(
-        sim.solver.local_elements
+        sim.solver.node_points, element_order
     )
     power = np.zeros(NUMBER_OF_TIME_STEPS)
     for time_step in range(NUMBER_OF_TIME_STEPS):
@@ -389,6 +425,7 @@ def test_thermo_piezo_time(tmp_path, test=True):
         theta[:, -1],
         nodes,
         elements,
+        element_order,
         sim.material_manager.get_heat_capacity(0),
         sim.material_manager.get_density(0)
     )
@@ -398,7 +435,7 @@ def test_thermo_piezo_time(tmp_path, test=True):
     # Maybe the signals are not fully dissipated?
     assert np.abs(input_energy - total_loss_energy) < 1e-11, \
         "Input energy does not equal total loss energy"
-    assert np.abs(total_loss_energy - stored_thermal_energy) < MAX_ERROR, \
+    assert np.abs(total_loss_energy - stored_thermal_energy) < ATOL, \
         "Total loss energy does not equal stored thermal energy"
 
     test_folder_name = "thermo_piezo_time"
@@ -419,19 +456,28 @@ def test_thermo_piezo_time(tmp_path, test=True):
         uut_u = sim.solver.u[:, -1]
         uut_mech_loss = sim.solver.mech_loss[:, -1]
 
-        print("Test q:", test_q)
-        print("UUT q:", uut_q)
-        print("Test u:", test_u)
-        print("UUT u:", uut_u)
-        print("Test mech loss", test_mech_loss)
-        print("UUT mech loss", uut_mech_loss)
-
         # Compare arrays
-        assert compare_numpy_array(uut_q, test_q), "Charge is not equal"
-        assert compare_numpy_array(uut_u, test_u), \
-            "Displacement u is not equal"
-        assert compare_numpy_array(uut_mech_loss, test_mech_loss), \
-            "Mech loss is not equal"
+        np.testing.assert_allclose(
+            uut_u,
+            test_u,
+            rtol=RTOL,
+            atol=ATOL,
+            err_msg="Displacement u is not equal"
+        )
+        np.testing.assert_allclose(
+            uut_q,
+            test_q,
+            rtol=RTOL,
+            atol=ATOL,
+            err_msg="Charge is not equal"
+        )
+        np.testing.assert_allclose(
+            uut_mech_loss,
+            test_mech_loss,
+            rtol=RTOL,
+            atol=ATOL,
+            err_msg="Charge is not equal"
+        )
     else:
         # Save data
         np.save(
diff --git a/uv.lock b/uv.lock
index 6ecec94..3f5a516 100644
--- a/uv.lock
+++ b/uv.lock
@@ -1,6 +1,6 @@
 version = 1
 revision = 2
-requires-python = ">=3.10"
+requires-python = ">=3.10, <3.13"
 resolution-markers = [
     "python_full_version >= '3.11'",
     "python_full_version < '3.11'",
@@ -78,19 +78,6 @@ wheels = [
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     { url = "https://files.pythonhosted.org/packages/37/a0/2410e5e6032a174c95e0806b1a6585eb21e12f445ebe239fac441995226a/charset_normalizer-3.4.2-cp312-cp312-win32.whl", hash = "sha256:db4c7bf0e07fc3b7d89ac2a5880a6a8062056801b83ff56d8464b70f65482b6c", size = 98357, upload-time = "2025-05-02T08:32:53.079Z" },
     { url = "https://files.pythonhosted.org/packages/6c/4f/c02d5c493967af3eda9c771ad4d2bbc8df6f99ddbeb37ceea6e8716a32bc/charset_normalizer-3.4.2-cp312-cp312-win_amd64.whl", hash = "sha256:5a9979887252a82fefd3d3ed2a8e3b937a7a809f65dcb1e068b090e165bbe99e", size = 105776, upload-time = "2025-05-02T08:32:54.573Z" },
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 [package.dev-dependencies]
 dev = [
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 [package.metadata.requires-dev]
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+dev = [
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