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)
- 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 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
04/12/2015: Advection-diffusion problems in 1D: numerical instability in the advection-dominated case
10/12/2015: Advection-diffusion problems in 1D: artificial diffusion
11/12/2015: Problem session
17/12/2015: Advection-diffusion problems in 2D: streamline diffusion; strongly consistent stabilization methods (SUPG, GALS, DWG)
18/12/2015: Stability analysis of GALS and convergence result
07/12/2015: Problem session; the heat equation
08/12/2015: The heat equation: semidiscretization in space, discretization in time
21/12/2015: The heat equation: stability analysis of the theta-method
22/12/2015: Remarks on the parabolic CFL condition and on error estimates; discontinuous Galerkin (DG) methods for the Poisson problem
28/12/2015: DG methods for advection-diffusion problems