SISSA course on Numerical Methods for PDEs
This is a Joint course between SISSA PhD in Mathematical Analysis, Modeling, and Applications, Laurea Magistrale in Matematica, and the Laurea Magistrale in Data Science and Scientific Computing.
Course materials are available on the course github repository.
Syllabus
Elliptic PDEs: boundary value problems, strong maximum principle, well-posedness.
Finite Difference (FD) Methods. Discrete Maximum Principle. Consistency, stability, convergence.
Basic notions on functional spaces, Weak formulations, Dirichlet principle, Lax-Milgram lemma. The method of Galerkin, Galerkin orthogonality, Cea Lemma. Finite Element Methods (FEM) for elliptic PDEs, implementation, conditioning, error analysis.
Interpolation, Bramble-Hilbert lemma. Generalised Galerkin method, truncation error and consistency, Strang Lemma. Convection-diffusion-reaction problems. The streamline diffusion method.
Initial and boundary value problems for parabolic PDEs, weak formulations, energy estimates, well posedness. FD and FEM discretisations.
Numerical methods for hyperbolic PDEs, method of characteristics, FD methods and the CFL condition. Upwind methods, Lax-Wendroff. Numerical dispersion. Leap-frog method. Discretisations of the wave equation.
Conservation laws. The Finite Volume (FV) method. FEM for hyperbolic problems.
Tools for Finite Element programming. Data structures and mesh generation, numerical quadrature techniques. Assembling and storage.
The following books are recommended:
- Larsson & Thomee Partial Differential Equations with Numerical Methods. Springer, 2009.
- Quarteroni Numerical Models for Differential Problems. Third edition. Springer, 2017.
- Morton & Mayers Numerical Solution of Partial Differential Equations. Cambridge, 1994.
Research Group: Mathematical Analysis, Modelling, and Applications
Location: SISSA main building room A-133
Next Lectures:
Tuesday, October 7, 2025 - 11:00 to 13:00
Wednesday, October 8, 2025 - 16:00 to 18:00
Tuesday, October 14, 2025 - 11:00 to 13:00
Wednesday, October 15, 2025 - 16:00 to 18:00
Tuesday, October 21, 2025 - 11:00 to 13:00
Wednesday, October 22, 2025 - 16:00 to 18:00
Tuesday, October 28, 2025 - 11:00 to 13:00
Wednesday, October 29, 2025 - 16:00 to 18:00
Tuesday, November 4, 2025 - 11:00 to 13:00
Wednesday, November 5, 2025 - 16:00 to 18:00
Tuesday, November 11, 2025 - 11:00 to 13:00
Wednesday, November 12, 2025 - 16:00 to 18:00
Tuesday, November 18, 2025 - 11:00 to 13:00
Wednesday, November 19, 2025 - 16:00 to 18:00
Tuesday, November 25, 2025 - 11:00 to 13:00
Wednesday, November 26, 2025 - 16:00 to 18:00
Tuesday, December 2, 2025 - 11:00 to 13:00
Wednesday, December 3, 2025 - 16:00 to 18:00
Tuesday, December 9, 2025 - 11:00 to 13:00
Tuesday, December 16, 2025 - 11:00 to 13:00
Wednesday, December 17, 2025 - 16:00 to 18:00