Numerical methods for partial differential equations

apl. Prof. Dr. Andreas M. Hinz

University of Munich, summer semester 2002

Contents

Partial differential equations play a key role in mathematical modelling of processes in science, technology and finance. Numerical methods to obtain practical solutions are therefore the core of what is now called scientific computing. They are based on discretization and depend on efficient algorithms for solving large algebraic systems. We will present two approaches, the finite difference method and the finite element method, as performed on the model problem of the Poisson equation.

Chapter 0. Introduction
1. Mathematical modelling
2. Analytic properties of partial differential equations
3. Discretization

Chapter 1. Finite Difference Method

4. Time-dependent problems
5. Finite Differences for the Poisson equation
6. Non-linear problems

Chapter 2. Finite Element Method

7. Approximation of solutions
8. Basic ideas of the Finite Element method
9. The Finite Element method for the Poisson equation

Prerequisites

Analysis, linear algebra. Some basic knowledge in numerical analysis and the analytic theory of partial differential equations is useful.

Literature

For the background in real analysis, numerical analysis and partial differential equations:
E. DiBenedetto, Real Analysis, Birkhäuser, Boston, 2002.
W. Gautschi, Numerical Analysis, Birkhäuser, Boston, 1997.
E. DiBenedetto, Partial Differential Equations, Birkhäuser, Boston, 1995.

A printed manuscript is available for the students of the course including an annotated list of references.

A. M. Hinz, andreas.hinz@mathematik.uni-muenchen.de, 2002-11-21