**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