NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS

Ph.D. Programme in Civil and Environmental Engineering, International cooperation and Mathematics (DICACIM)
(A.Y. 2023/24)

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
06/05/2024. 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
08/05/2024. 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.	3-LG-quadrature.pdf
Lesson 3
10/05/2024. 9.00-12.00. MATLAB/Lecture	Numerical solution of 1D elliptic problems with Finite Elements and MATLAB. Error analysis. P2-fem.	FEM1d.zip 4-lab-fem1.pdf 5-1d-fem-error.pdf 6-linearsystems.pdf
Lesson 4
13/05/2024. 9.00-12.00. Lecture	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. Construction of the linear system. Error estimates. Generalization to higher polynomial degree. A short review of Spectral Element Methods.	7-2dfem.pdf 8-sem.pdf
Lesson 5
15/05/2024. 9.00-12.00. MATLAB	Numerical solution of 2d elliptic equations with MATLAB pdetool	9-lab-fem2d.pdf FEM2d.zip
Lesson 6
16/05/2024. 9.00-12.00. Lecture/MATLAB	Parabolic problems: weak formulation. Approximation in space by FEM and in time by finite differences. Stability and Convergence analysis. Numerical solution of Parabolic problems with MATLAB.	FEM1d.zip 10-lab-parabolic.pdf
Lesson 7
20/05/2024. 9.00-12.00. Lecture/MATLAB	Advection diffusion reaction problems. Artificial diffusion, stramline diffusion, and stabilization techniques. Numerical solution of advection-diffusion problems.	FEM1d.zip FEM2d.zip 11-lab-ad.pdf

Paola Gervasio   -   May 2024