## Adaptive Techniques in Numerical Analysis

- Nov. 6, 2001
- 3:30 p.m.
- LeConte 312

## Abstract

Dynamically adapting discretizations in the course of a computational solution process is a powerful concept for reducing the computational complexity of large scale numerical simulation tasks. The promising potential of adaptivity has stimulated numerous investigations from several different perspectives, ranging from abstract complexity theory to extensive experimental numerical studies. While computational experience appears to confirm high expectations (in some contrast to certain theoretical predictions) a rigorous assessment of the performance of adaptive methods has still remained far from being satisfactory. A central problem is to derive error and complexity estimates relating the computational work and the adaptively generated number of degrees of freedom to the achieved accuracy. This lecture is devoted to recent progress concerning this issue for a large class of variational problems. It will be indicated how some tools from nonlinear approximation and harmonic analysis suggest on one hand new algorithmic paradigms in the context of wavelet schemes while, on the other hand, they lead to first optimal error and complexity estimates in the context of finite element discretizations. Some numerical illustrations are presented.