Numerical Methods for Partial Differential Equations
University of Vienna, Wintersemester 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: Thu 11.30-13.00 SR08, Fri 11.30-13.00 SR08, Oskar-Morgenstern-Platz 1
First Lecture: Thu, October 8, 2015

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 programming activities, 50% Final exam

Program:
08/10/2015: Course introduction; elements of functional spaces: spaces of continuous functions
09/10/2015: Elements of functional spaces: spaces of integrable functions, distributions and distributional derivatives
15/10/2015: Elements of functional spaces: Sobolev spaces; Poincaré-Friedrichs inequality, some Sobolev embeddings, traces
16/10/2015: Variational formulation of elliptic problems, the Lax-Milgram lemma
22/10/2015: Galerkin methods: definition, matrix form, well-posedness, Céa's lemma and convergence
23/10/2015: Generalized Galerkin methods: Strangs' lemma; problem session
29/10/2015: Introduction to Finite Element Methods (FEM): meshes, spaces of piecewise polynomials
30/10/2015: Introduction to FEM: basis functions and degrees of freedom; problem session
05/11/2015: Interpolation operators; FEM approximation of the Poisson problem: computing local matrices and rhs
06/11/2015: FEM approximation of the Poisson problem: assembling of global matrices and rhs, imposition of Dirichlet boundary conditions
12/11/2015: Interpolation operators; Matlab implementation of P1-FEM for the Poisson problem with Dirichlet b.c.
13/11/2015: Problem session
19/11/2015: Interpolation errors: Bramble-Hilbert's lemma and Deny-Lions' lemma
20/11/2015: Bounds of the local and of the global interpolation errors
26/11/2015: Estimate of the FEM error in the L2-norm (duality argument)
27/11/2015: Complements: inverse estimate; FEM approximation of the Hemlholtz problem: matrix form
03/12/2015: FEM approximation of the Hemlholtz problem: stability and error analysis