**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