215 IGPM215.pdf        May 2002
TITLE Adaptive Wavelet Methods Basic Concepts and Applications to the Stokes Problem
AUTHORS Wolfgang Dahmen, Jürgen Vorloeper, Karsten Urban
ABSTRACT This paper is concerned with recent developments of adaptive wavelet schemes. Central issues are the design of such algorithms and concepts for proving their asymptotically optimal complexity properties when compared with best N-term approximation. After describing the scope of variational problems to be treated, the main concepts of adaptive strategies are brie reviewed. This is subsequently applied to the important class of saddle point problems. A new convergence proof exemplifies the basic ingredients of the complexity analysis. Finally, the theoretical results are applied to the Stokes problem as a representative example of saddle point problems. In particular, we propose a new variant of an Uzawa iteration based on a different treatment of the divergence operator. We conclude with some numerical comparisons of the different versions of the adaptive saddle point schemes.
KEYWORDS Variational problems, saddle point problems, wavelet bases, adaptive application of operators, Stokes equations, convergence estimates
PUBLICATION Wavelet Analysis -- Twenty Years Developments, Ding--Xuan Zhou ed., World Scientific, New Jersey,
2002, 39--80