NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS

Ph.D. Programme in Civil and Environmental Engineering, International cooperation and Mathematics (DICACIM)
(A.Y. 2024/25)

Programme

• Elliptic equations:
  • approximation via the Galerkin method;
  • the Finite Element Method;
  • the Spectral Element Method;
  • error estimates.
• Parabolic equations:
  • the semi-discrete problem, convergence analysis;
  • the theta-method for time discretization: stability and convergence analysis.
• Diffusion-transport-reaction equations:
  • stabilization methods.

MATLAB campus license

Textbooks

A. Quarteroni, Numerical Models for Differential Problems (free download for Unibs users).
A. Quarteroni, F. Saleri, P. Gervasio Calcolo Scientifico (free download for Unibs users).
A. Quarteroni, F. Saleri, P. Gervasio Scientific Computing.

Lesson 1
28/04/2025 9.00-12.00. Lecture	Introduction to PDEs. Weak derivatives, the spaces L2 and H1. 1d case: elliptic problems with Dirichlet, Neumann or Robin b.c. and their weak forms. Abstract setting: the Lax-Milgram lemma.	0-summary.pdf 1-intro.pdf 2-fun-analysis.pdf
Lesson 2
29/04/2025. 9.00-12.00. Lecture	1d case: elliptic problems with non-homogeneous Dirichlet conditions. Galerkin approximation, Céa Lemma, stability and convergence. P1-fem: Lagrangian basis, matrix and right hand side assembling. Legendre-Gauss quadrature formulas.
Lesson 3
30/04/2025. 9.00-12.00. MATLAB/Lecture	Numerical solution of 1D elliptic problems with Finite Elements and MATLAB. Error analysis. P2-fem.
Lesson 4
05/05/2025. 9.00-12.00. Lecture
Lesson 5
06/05/2025. 9.00-12.00. MATLAB
Lesson 6
08/05/2025. 9.00-12.00. Lecture/MATLAB
Lesson 7
09/05/2025. 9.00-12.00. Lecture/MATLAB

Paola Gervasio   -   April 2025