Wavelet methods for PDEs — some recent developments

This paper is concerned with recent developments of wavelet schemes for the numerical treatment of operator equations with special emphasis on two issues: adaptive solution concepts and nontrivial domain geometries. After describing a general multiresolution framework the key features of wavelet bases are highlighted, namely locality, norm equivalences and cancellation properties. Assuming first that wavelet bases with these properties are available on the relevant problem domains, the relevance of these features for a wide class of stationary problems is explained in subsequent sections. The main issues are preconditioning and the efficient (adaptive) application of wavelet representations of the involved operators. We indicate then how these ingredients combined with concepts from nonlinear or best N-term approximation culminate in an adaptive wavelet scheme for elliptic selfadjoint problems covering boundary value problems as well as boundary integral equations. These schemes can be shown to exhibit convergence rates that are in a certain sense asymptotically optimal. We conclude this section with some brief remarks on data structures and implementation, interrelations with regularity in a certain scale of Besov spaces and strategies of extending such schemes to unsymmetric or indefinite problems. We address then the adaptive evaluation of nonlinear functionals of wavelet expansions as a central task arising in connection with nonlinear problems. Wavelet bases on nontrivial domains are discussed next. The main issues are the development of Fourier free construction principles and criteria for the validity of norm equivalences. Finally, we indicate possible combinations of wavelet concepts with conventional discretizations such as finite element or finite volume schemes in connection with convection dominated and hyperbolic problems.