Numerical Methods for Elliptic and Parabolic Partial Differential Equations Autumn 2024

Lecturer: Jörg Nick
Coordinator: Thea Kosche

Lectures

Day	Time	Room
Tuesday	10:15-11:45	HG D 3.2
Wednesday	12:15-14:00	LFW B 1

On tuesdays, the lecture will not have a break at 11:00, however will finish early. This way there is some time to have a break before the exercise class at 13:15.

Content

Partial differental equations
Sobolev spaces
Variational problems
Weak solutions
One-dimensional linear finite elements
Simplicial finite elments in d dimensions
Implementation
A posteriori error estimates and adaptive grid refinement
Numerical solutions of the discrete problem
Parabolic PDEs

Lecture notes

The main notes used for this lecture will be the lecture notes written by Prof. Sauter.

Lecture Content

Week	Date	Topics
Week 1	September 17	Classification of linear PDEs Starting chapter of Sobolev spaces Definition of weak derivative
Week 1	September 18	Definition Sobolev spaces Connection of weak derivatives and piecewise derivatives
Week 2	September 24	Definition of Dirichlet trace Trace spaces Trace theorem
Week 2	September 25	Weak definition of Neumann trace Friedrich's inequality Motivation: Variational problems
Week 3	October 1	Introduction: Abstract variational problems Lax-Milgram Lemma Trace theorem
Week 3	October 2	Cea's Lemma Aubin-Nitsche trick
Week 4	October 8	Weak formulation Existence and uniqueness of weak solutions
Week 4	October 9	Higher regularity considerations One dimensional linear finite elements
Week 5	October 15	L^2-estimates (Aubin-Nitsche trick) Formulation as linear system
Week 5	October 16	Finite elements in 2D Grids via reference element
Week 6	October 22	Simplices Nodes and polynomials
Week 6	October 23	Nodes and polynomials Functional analytic tools (compact embeddings, Sobolev embeddings)
Week 7	October 29	Transformation of interpolation error to reference element
Week 7	October 30	Error estimates for finite element methods
Week 8	November 5	Implementation of 2D FEM I
Week 8	November 6	Implementation of 2D FEM II Lecture codes
Week 9	November 12	Grid refinements A posteriori error analysis I
Week 9	November 13	A posteriori error analysis II

Exercises

The exercises will be posted here. There will be no solutions provided, however some solutions will be discussed during the exercise class.

Exercise sheet	Exercise class
Exercise sheet 1	September 24
Exercise sheet 2	October 1 (online, zoom link will be sent via e-mail)
Exercise sheet 3	October 8
Exercise sheet 4, Template, Coding Solution	October 15
Exercise sheet 5, Coding solution	October 22
Exercise sheet 6, Code	November 12
Exercise sheet 7, TriangulationMeshes	November 19

Exercise class

The exercise class will start in the second week of the semester.

Day	Time	Room
Tuesday	12:15-13:00	HG E 33.5

Literature

The Mathematical Theory of Finite Element Methods by Susanne C. Brenner, L. Ridgway Scott (Springer 1994)
Numerical Approximation of Partial Differential Equations by Sören Bartels (Springer 2016)