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

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
Week 9 November 12
  • Grid refinements
  • A posteriori error analysis: Reliability I
Week 9 November 13
  • A posteriori error analysis: Reliability II
Week 10 November 19
  • A posteriori error analysis: Reliability III
Week 10 November 20
  • A posteriori error analysis: Efficiency I
Week 11 November 27
  • A posteriori error analysis: Efficiency II
Week 11 November 28
  • A posteriori error analysis: Efficiency III
  • Adaptive methods
Week 12 December 3
  • Introduction to parabolic PDEs
  • Abstract evolution problems
Week 12 December 4
  • The exponential operator
  • Formulation of the spatially discrete problem as an abstract evolution problem

Exercises

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

Exercise sheetExercise 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 sheet 8, fem_tools.py, grid_tools.py, solutions.py November 26
Exercise sheet 9,Coding Example 9 December 3 and December 10
Exercise sheet 10 December 17

Exercise class

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

DayTimeRoom
Tuesday12:15-13:00HG E 33.5

Literature