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 blackboard
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 for the Poisson problem with Neumann bc. Legendre-Gauss quadrature formulas.	blackboard 3-LG-quadrature.pdf
Lesson 3
30/04/2025. 9.00-12.00. MATLAB/Lecture	P1-fem: matrix and right hand side assembling for the Poisson problem with non-homogeneous Dirichlet bc. Coding in MATLAB. Numerical solution of 1D elliptic problems with P1-fem.	FEM1d.zip 4-lab-fem1.pdf
Lesson 4
05/05/2025. 9.00-12.00. Lecture/MATLAB	Error analysis for P1-fem.	4-lab-fem1.pdf 5-1d-fem-error.pdf
Lesson 5
06/05/2025. 9.00-12.00. Lecture/MATLAB	P2-fem. Error analysis. Weak form of elliptic problems with Dirichlet, Neumann, and mixed b.c. in 2d/3d. Galerkin formulation. Finite Elements discretization. Triangulations. P1 and Q1 FEM.	4-lab-fem1.pdf 5-1d-fem-error.pdf 6-P2fem.pdf 7-2dfem.pdf
Lesson 6
08/05/2025. 9.00-12.00. Lecture/MATLAB	P1-fem approximation of 2d elliptic problems: construction of the linear system. Numerical solution of 2d elliptic equations with MATLAB pdetool. Parabolic problems.	8-lab-fem2d.pdf 9-linearsystems.pdf
Lesson 7
09/05/2025. 9.00-12.00. Lecture/MATLAB	Parabolic problems. Approximation in space by FEM and in time by finite differences. Stability and Convergence analysis. Numerical solution of Parabolic problems with MATLAB.	FEM1d_heat.zip 10-lab-parabolic.pdf 11-pdeModeler.pdf

Paola Gervasio   -   April 2025