Space-time methods
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WS 2023/2024
Lecturer: Wick/Fischer/Roth
Leibniz University Hannover, Institute of Applied Mathematics
Format: 2+1 (90 minutes lecture per week; 45 minutes exercise per week)
Times: Thu 17:15 - 18:45
Place: Room C311
First lecture: Oct 12, 2023
Last lecture: Jan 25, 2024
Summary
=======
This lecture is devoted to space-time finite element methods. First, we concentrate on formulations,
numerical modeling and their discretization. Therein, the focus is on space-time
finite element discretizations with discontinuous Galerkin (dG) methods in time
and continuous Galerkin (cG) schemes in space. The governing concepts of the
numerical solution will be introduced as well. Afterward, we dive into concepts
of goal-oriented error control and adaptivity. We finish with current developments
of space-time goal-oriented error-controlled adaptivity applied to model order reduction.
Note
====
The main focus of this course is on the computer-based numerical solution of
time-dependent partial differential equations in space-time. Prototypical applications
include the heat, wave, elastodynamics or Navier-Stokes equations.
Please do not confuse this course with a physics lecture on special or general relativity.
Main literature
===============
[1] T. Wick; Space-time methods, book, to appear, 2023; current version
https://thomaswick.org/links/Wi23_st_book_preprint_Aug_8_2023.pdf
[2] H. Fischer, J. Roth, T. Wick, L. Chamoin, A. Fau;
MORe DWR: Space-time goal-oriented error control for incremental POD-based ROM;
arXiv 2023, https://arxiv.org/abs/2304.01140
Plan
====
A) Introduction to space-time
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1. Background and motivation of this lecture
Notation, space-time Sobolev spaces, Bochner spaces
Heat equation in classical and space-time form
2. ODEs in space-time
Time-stepping schemes
3. PDEs in space-time form: discretization in space and time
4. Space-time solution (I): nonlinear solution
5. Space-time solution (II): linear solution
B) Error estimation and adaptivity part for ODEs
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6. Goal-oriented error estimation ODEs
Introduction via Ax=b
7. Goal-oriented error estimation ODEs
Final adaptive algorithm
C) Error estimation and adaptivity part for PDEs
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8. Space-time adaptivity for PDEs
Optimization problems, Lagrange formalism, goal functionals
9. Discussions of adjoint
(linear, running backwards in time, partial integration in time)
10. Error control and localization
11. Adaptive algorithm (chapter 12),
road map to model problems (Chapter 12.3)
D) Advanced techniques and current state-of-the-art
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12. MORe DWR: Model order reduction with DWR (I)
Lit.: https://arxiv.org/abs/2304.01140
13. MORe DWR: Model order reduction with DWR (II)
Lit.: https://arxiv.org/abs/2304.01140
14. Quiz
THE END
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