# Adaptive Lösungskonzepte/Adaptive solution concepts (SoSe 2023)

Prof. Dr. Markus Bachmayr

Manfred Faldum, M.Sc.

## Event |
## Time |
## Location |
## SWS |
---|---|---|---|

## Lecture |
Monday, 14:30 - 16:00 Friday, 10:30 - 12:00 |
Main building, room 149 Main building, room 149 |
4 SWS |

## Exercise |
Friday, 12:30 - 14:00 | Main building, room 149 | 2 SWS |

### Contents

A central motivating question is the following: given a partial differential equation, how can we find approximate solutions for each given error tolerance that can be represented by a near-minimal number of coefficients, without using any explicit knowledge on the respective exact solution? This naturally leads to iterative procedures that perform a gradual refinement of discretizations. Ideally, the computational costs (in terms of the number of elementary operations) required by such methods should scale linearly with respect to the number of coefficients in the produced approximate solutions. In the lecture, we will see in particular how methods that provably have this optimality property can be constructed. The course outline will be as follows:- Fundamentals of nonlinear approximation and adaptive basis expansions
- Stable variational formulations of operator equations
- Adaptive wavelet methods for differential equations and integral equations
- Adaptive finite element methods
- Adaptivity for higher-dimensional problems

**Prerequisites:** Numerik 1-4, in particular basic knowledge on numerics of elliptic partial differential equations

### News and information

- The lecture and exercises will be held in English or German according to demand.
- Lecture notes (in English) will be provided.
- Topics for the master seminar Aktuelle Themen der Numerik (SoSe 23) are available that can be combined well with the lecture. In case of interest, please contact Prof. Bachmayr directly.

### Registration

Registration is via RWTHonline. If you register for the tutorial (Übung) you will automatically get access to the course room in RWTHmoodle.

### Literature

Some selected articles (more literature references will be given in the lecture):

- J. M. Cascon, C. Kreuzer, R. H. Nochetto, and K. G. Siebert.
*Quasi-optimal convergence rate for an adaptive finite element method*, SIAM J. Numer. Anal. 46 (2008). - A. Cohen, W. Dahmen, and R. DeVore.
*Adaptive wavelet methods for elliptic operator equations: convergence rates*, Mathematics of Computation 70 (2001). - T. Gantumur, H. Harbrecht, and R. Stevenson.
*An optimal adaptive wavelet method without coarsening of the iterands*, Math. Comp. 76 (2007).