Numerical Methods for Partial Differential
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
- 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.
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
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
21/05/2015: Error analysis of GALS
28/05/2015: The heat equation
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)