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)
- 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.
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
05/05/2015: Advection-diffusion problems in 1D: numerical instability in the advection-dominated case
07/05/2015: Advection-diffusion problems in 1D: artificial diffusion; 2D case: streamline diffusion
12/05/2015: Strongly consistent stabilization methods for advection-diffusion problems (SUPG, GALS, DWG); stability analysis of GALS
19/05/2015: Laboratory
21/05/2015: Error analysis of GALS
28/05/2015: The heat equation
02/06/2015: Laboratory
09/06/2015: The heat equation: semidiscretization in space
11/06/2015: The heat equation: discretization in time and stability analysis of the theta-method
16/06/2015: PDE Workshop
18/06/2015: Stability analysis of the theta-method (conclusion); complements: conditioning of the FE mass matrix, inverse trace inequality
23/06/2015: Discontinuous Galerkin (DG) approximation of advection-diffusion problems: interior penalty (IP) methods for the diffusion, upwind for the advection
25/06/2015: Analysis of IP-DG approximations of the Poisson problem (sketch)