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Foundations of the theory of partial differential equations relevant in engineering applications and their numerical solution.

### Prerequisites

The course links topics well known from bachelor mathematics courses and extends them, in particular linear algebra, analysis and numerical mathematics. Expected competences:

Linear algebra: systems of equations, matrices, numerical methods

Analysis: partial derivatives, gradient, concept of ordinary differential equation, linear differential equations, separable differential equation, concept of fourier series.

### Learning Objectives

The student knows the basic geometric, analytic and numeric aspects of partial differential equations. He/she knows the basic methods to successfully solve partial differential equations analytically and numerically as well as a set of typical examples that allow to better understand the theoretical concepts.

### Contents of Module

Part 1: General theory

Goals of part 1:

- understand how partial differential equations naturally appear in applications
- be able to solve selected examples using the separation method
- understand the kinds of boundary conditions necessary, Dirichlet and Neumann boundary conditions
- create a collection of examples to illustrate the basic theoretical principles

Lesson plan for part 1:

- From ordinary to partial differential equations: three applied examples: wave equation, Laplace equation and heat equation. Goal: understand how partial differential equations naturally appear in applications
- Quasilinear partial differential equations of first order, solutions using characteristics.
- Solution of partial differential equations using separation of variables.
- Solution of partial differential equations using the Laplace- or Fourier-transforms.
- Elliptic partial differential equation with the Laplace equation as the prime example. Poisson formula, maximum principle, uniqueness of solutions.
- Parabolic differential equations with the heat equation as prime example. Maximum principle and kernel function, Green's function.
- Hyperbolic partial differential equations with the wave equation as prime example. d'Alembert solution and method of characteristics.

Part 2: Numerical methods for partial differential equations

- Analysis of finite difference methods in the example of the two point boundary problem:
- condition
- stability
- convergence

Goal: understand some central ideas and concepts of the analysis of numerical methods in general and finite difference methods in particular. - Finite volume methods for the Poisson-equation:
- Example of a cell centered finite volume difference method
- Example of a node centered finite volume element method
- Some examples regarding Navier-Stokes

Goal: construct a collection of numerical methods that represent the broadness of possible approximation techniques. - Finite element method in the example of the stationary heat equation:
- differential, variational and integral formulation
- global and local ansatz functions
- elements and element types

- General perspective: weighted residues.

Goal: concise introduction into the methodology of finite elements. - Problems of finite element methods in the example of the beam equation. Solution strategies and their numerical background:
- p-strategies
- h-strategies
- r-strategies

Goal: show limitations of finite element methods - Finite elements in the example of the nonstationary heat equation:
- semidiscrete schemata
- completely discrete schemata

- Eigenvalue determination using finite elements in the example of the beam oscillation equation.

Goal: illustrate additional fields of application for finite elements.

This module does not intend to teach the use of any particular software product for the solution of partial differential equations. Instead it strives to teach the foundations for their successful use. The students should become capable to judge the potential and limitations of such a software system und the precision and reliability of the results that can be expected from such a system.

### Literature

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