**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)