Exercise Sheets

I. Introduction

1. Classification, Transformation of Variables (pdf)

II. Difference Methods for parabolic Differential Equations

2. Difference Approximation, Order of Consistency (pdf)
(Lab-Exercise: Link: Solution of the Heat Equation,

Matlab-File: ueb2.m)
3. l²-Stability in the Sense of von Neumann (pdf)
4. Tridiagonal Matrices, Discrete Maximum Principle, Stability in Maximum Norm (pdf)

III. Difference Methods for elliptic Differential Equations

5. Poisson Equation, Maximum Principle (pdf)
(Lab-Exercise: Link: Solution of the Helmholtz Equation,

Matlab-File: ueb5.m)
6. Lab-Exercise: Poisson Equation (pdf)
7. Discretizations of higher Order (pdf)

IV. Introduction to the Theory of Sobolev Spaces

8. Distributive Derivative, Sobolev Spaces, Dual Spaces (pdf)
9. Inequality of Friedrichs, Inequality of Poincaré, Sobolev Inequality (pdf)

V. Variational Formulations of Boundary Value Problems

10. Weak Formulation, Lemma of Lax-Milgram (pdf)

VI. The Finite Element Method

11. Stiffness Matrix, Triangulation (pdf)
12. Linear Finite Elements, Transformationformula, inverse Inequality (pdf)

VII. Introduction to Multigrid Methods

13. The Method of simple Iteration, V-Cycle, W-Cycle, Remainder Projector (pdf)

Information Resources for Matlab and Scilab

Matlab PDE Toolbox

The Partial Differential Equation Toolbox

Eine sehr kurze Einführung in die Partial Differential Equation Toolbox von Matlab

Using the Matlab PDE toolbox

The PDE Toolbox - Minicourse

Finite Elemente

Das kleine Finite-Elemente-Skript, VL-Skript von A. Jüngel, 2004.

Finite Elemente in Scilab: Das Lösen partieller Differentialgleichungen mit Hilfe der FreeFEM-Toolbox

FreeFEM for Scilab / SciFreeFEM

Openfem_scilab

OpenFEM (An Opensource Finite Element Toolbox)

CALFEM (Computer Aided Learning of the Finite Element Method)

Korrekturen zum Buch von Braess

Mehrgitterverfahren

MGNet - Homepage

Multigrid Workbench (self-guided tour through multigrid algorithms)

An Introduction to Multigrid Methods (Online-Buch von Pieter Wessling)

MGLab (interactive environment for experimenting with multigrid algorithms, MATLAB)

A short paper describing MGLab

MUDPACK multigrid software package (fortran)

• Lloyd N. Trefethen, Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations, unpublished text, 1996, (now available online).
• Some disasters attributable to bad numerical computing (Douglas N.Arnold)
• PDELab: Problem Solving Environment for PDE Applications (Purdue University)
• NumaWWW : Numerical Mathematics in the WorldWideWeb (TU Darmstadt):
• Partial Differential Equations