Numerical Methods for Partial Differential Equations
University of Vienna, Sommersemester 2015

The course mainly focuses on Finite Element Methods (FEM) for the numerical approximation of Partial Differential Equations.

Aim: presenting theoretical and numerical aspects of FEM for the numerical approximation of Partial Differential Equations arising from different applications, from theoretical stability and error analysis, to implementation.

Schedule: Tue 09.45-11.15 SR10, Thu 08.00-09.30 SR10, Oskar-Morgenstern-Platz 1
First Lecture: Tue March 3, 09.45 SR10

Outline:
- Functional Analysis tools (recap)
- FEM for the Poisson problem (stability, error analysis, and implementation)
- FEM for the heat equation (coupling space and time discretization)
- FEM for the Helmholtz problem (analysis of indefinite problems by duality argument)
- Depending on the students' interests, we will deal with either other applications (fluid mechanics, electromagnetism, elasticity), or non standard FEM (as discontinuous Galerkin FEM), or domain decomposition techniques.

A. Quarteroni, Numerical Models for Differential Problems, Springer, 2014.
A. Quarteroni, A. Valli, Numerical Approximation of Partial Differential Equations, Springer, 1994
Other material will be distributed during the course.

Evaluation: 50% Homework and Lab activities, 50% Final exam

Program:
03/03/2015: Elements of functional spaces: spaces of continuous functions, spaces of integrable functions
05/03/2015: Elements of functional spaces: Distributions and distributional derivatives, Sobolev spaces, Poincaré-Friedrichs inequality, some Sobolev embeddings.
10/03/2015: Traces. Variational formulation of elliptic problems, the Lax-Milgram lemma
12/03/2015: Rektorstag (no class)
17/03/2015: Variational formulation of elliptic problems: the symmetric case. Galerkin methods: definition, matrix form, well-posedness
19/03/2015: Céa's lemma and convergence; generalized Galerkin methods: Strangs' lemma
24/03/2015: Introduction to Finite Element Methods (FEM): meshes, spaces of piecewise polynomials
26/03/2015: Introduction to FEM: basis functions and degrees of freedom; interpolation operators
14/04/2015: FEM approximation of the Poisson problem: computing local matrices and rhs; assembling of global matrices and rhs
16/04/2015: FEM approximation of the Poisson problem:imposition of Dirichlet boundary conditions; interpolation errors
21/04/2015: Bramble-Hilbert's lemma and Deny-Lions' lemma; bounds of the local and of the global interpolation errors
23/04/2015: Estimate of the FEM error in the L2-norm (duality argument)
28/04/2015: Complements: conditioning of the stiffness matrix for coercive elliptic problems
30/04/2015: FEM approximation of the Hemlholtz problem