Numerical Methods for Partial Differential Equations
University of Vienna, Sommersemester 2014

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 8:30-10:00 SR10, Wed 11:15-12:45 SR10 (only on Wed 05/03/2014: 13:15-14:45 SR09), Oskar-Morgenstern-Platz 1
First Lecture: Tue March 4, 2014

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)
- FEM for advection-dominated advection-diffusion problems (stabilization mechanisms)
- 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.

Suggested reading: A. Quarteroni, Numerical Models for Differential Problems, Springer, 2014.
Other material will be distributed during the course.

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

Program:
04/03/2014: Elements of functional spaces: spaces of continuous functions, spaces of integrable functions
05/03/2014: Elements of functional spaces: Distributions and distributional derivatives, Sobolev spaces, Poincaré-Friedrichs inequality, some Sobolev embeddings, traces.
11/03/2014: Variational formulation of elliptic problems, the Lax-Milgram lemma
18/03/2014: Galerkin methods: definition, well-posedness and convergence (Céa's lemma), matrix form
19/03/2014: Generalized Galerkin methods: Strangs' lemma; introduction to Finite Element Methods (FEM): meshes
25/03/2014: Lab 1
26/03/2014: Introduction to FEM: spaces of piecewise polynomials, basis functions and degrees of freedom
01/04/2014: Interpolation operators; FEM approximation of the Poisson problem: computing local matrices and rhs
02/04/2014: FEM approximation of the Poisson problem: assembling of global matrices and rhs; imposition of Dirichlet boundary conditions; interpolation errors
08/04/2014: Lab 2
09/04/2014: Bramble-Hilbert's lemma and Deny-Lions' lemma
29/04/2014: Bounds of the local and of the global interpolation error; estimate of the FEM error in the H1-norm
30/04/2014: Estimate of the FEM error in the L2-norm (duality argument); FEM approximation of the Hemlholtz problem
06/05/2014: Lab 3
13/05/2014: Error estimates for FEM approximation of the Helmholtz problem; complements: conditioning of the stiffness matrix for coercive elliptic problems
14/05/2014: Conditioning of the stiffness matrix: inverse inequality and conclusion; the heat equation
20/05/2014: Lab 4
21/05/2014: The heat equation: semidiscretization in space; discretization in time
27/05/2014: Stability analysis of the theta-method
28/05/2014: Advection-diffusion problem in 1D: numerical instability
03/06/2014: Lab 5
04/06/2014: No class
11/06/2014: Advection-diffusion problem in 1D: artificial diffusion; 2D case: streamline diffusion
17/06/2014: Strongly consistent stabilization methods for advection-diffusion problems (SUPG, GALS, DWG); stability analysis of GALS
18/06/2014: Error analysis of GALS