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 at the course GitHub repo https://github.com/andreacangiani/NMPDE2026.
The course provides a high level overview on the numerical solution of Partial Differential Equations (PDEs), covering mathematical modelling, functional analysis setting, discretization via classical approaches such as Finite Differences (FD) and Finite Element Methods (FEM), and Finite Volume methods, numerical analysis, computational implementation.
A range of PDE problems will be presented, covering different types of PDEs (elliptic, parabolic, hyperbolic) both linear and non-linear. Each problem is analyzed during the lectures and implemented at the computer practicals. Key aspects such as consistency, stability, and convergence will be thoroughly investigated, providing the guidelines for the correct choice and implementation of numerical methods for a range of problems.
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.
Lecture notes will be provided. 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.