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12163-HS2023-0-Partial Differential Equations





Root number 12163
Semester HS2023
Type of course Lecture
Allocation to subject Mathematics
Type of exam Written exam
Title Partial Differential Equations
Description Course contents

1 Introduction
1.1 Differential operators
1.2 Partial differential equations: Some examples
1.3 Well-posedness
1.4 Characteristics, D’Alembert’s formula for the wave equation

2 Energy Methods
2.1 The Dirichlet Principle
2.2 Stability estimates and energy properties
2.2.1 Stability of Poisson problem
2.2.2 HeatEquation
2.2.3 HomogeneousWaveEquation
2.3 UniquenessofSolutions

3 Classical Solutions of the Poisson Equation
3.1 The Poisson equation in Rm
3.1.1 Harmonic functions
3.1.2 Fundamental solution
3.1.3 Solution formula in Rm
3.2 The Poisson equation on a bounded domain
3.2.1 Green functions
3.2.2 Application: Poisson’s formula for the sphere
3.2.3 Maximum principle

4 Heat Equation
4.1 Homogeneous Heat Equation
4.1.1 A Random Walk Model
4.1.2 Heat Kernel
4.1.3 Solution of the Homogeneous Heat Equation
4.1.4 Maximum Principles
4.2 Inhomogeneous Heat Equation and Duhamel’s Principle

5 Weak Solutions of the Poisson Equation
5.1 Abstract Hilbert space theory
5.1.1 Orthogonality and the projection theorem
5.1.2 Linear forms and the Riesz representation theorem
5.1.3 Bilinear forms and weak formulations
5.1.3.1 Symmetric problems and the Dirichlet principle
5.1.4 The Lax-Milgram Theorem
5.1.5 Abstract Galerkin discretizations
5.1.5.1 Discrete weak formulation
5.1.5.2 Quasi-optimality and convergence
5.2 Application to the Poisson problem
5.2.1 Sobolev spaces
5.2.1.1 Weak derivatives
5.2.1.2 The Sobolev spaces W1,2(Ω) and H01(Ω)
5.2.1.3 Poincaré-Friedrichs inequalities
5.2.1.4 Traces
5.2.2 Weak solution of the Poisson equation
5.2.3 Galerkin approximation of the Poisson equation
5.2.3.1 Triangulations
5.2.3.2 P1-finite element spaces
5.2.3.3 P1-finite element formulation
5.2.3.4 Practical solution
5.2.4 More general partial differential equations
5.3.2 The Poisson problem with Neumann boundary data

6 Nonlinear monotone PDE
6.1 Contractions
6.1.1 Zarantonello’s theorem
6.1.2 Application to strongly monotone PDE
6.2 Potentials
6.2.1 Gâteaux differentiability
6.2.2 Convergence of the Kacanov iteration
6.2.3 Application to strongly monotone PDE
ILIAS-Link (Learning resource for course) Registrations are transmitted from CTS to ILIAS (no admission in ILIAS possible). ILIAS
Link to another web site
Lecturers Prof. Dr. Thomas Wihler, Institute of Mathematics ✉
ECTS 6
Recognition as optional course possible Yes
Grading 1 to 6
 
Dates Monday 15:15-17:00 Weekly
Tuesday 08:15-10:00 Weekly
 
Rooms Hörraum 119, Exakte Wissenschaften, ExWi
 
Students please consult the detailed view for complete information on dates, rooms and planned podcasts.