Wavelet Methods for Second-Order Elliptic Problems, Preconditioning, and Adaptivity

Wavelet methods allow to combine high order accuracy, eecient preconditioning techniques and adaptive approximation, in order to solve eeciently elliptic operator equations. Many diiculties remain, in particular related to the adaptation of wavelet decompositions to bounded domains with prescribed boundary conditions, as well as the possibly high constants in the O(1) preconditioning. In this paper we consider second order operators on tensor product domains. For such domains, we discuss the construction of high order multiresolution approximation and wavelet bases, and in particular the choice of the wavelets near the boundary in order to optimize the eeciency of the diagonal preconditioning of elliptic operators. In order to improve the constants obtained by such simple diagonal preconditioning, we propose an almost diagonal preconditioner based on solving local Petrov-Galerkin problems. The eeciency of this method is illustrated by solving elliptic second order problems with variable or constant coeecient and homogeneous boundary conditions on a uniform discretization. Finally, we propose a coupling of the iterative solver with an adaptive space reenement technique. On the Laplacian model problem, our experiments show that this algorithm generates an optimal nonlinear approximation of the solution. Both isotropic and anisotropic decompositions are considered and compared in terms of preconditioning and compression of the solution. R esum e: Pour la r esolution d' equations elliptiques, les m ethodes d'ondelettes permettent d'obtenir a la fois des approximations d'ordre elev e, des techniques de pr econditionnement et des strat egies d'approximations adaptatives simples et eecaces. De nombreuses diicult es de mise en oeuvre de ces m ethodes r esident tou-jours dans la d eenition de bases d'ondelettes avec conditions aux limites sur des domaines g en eraux, ainsi que dans l'optimisation des constantes du pr econditionnement O(1). Dans cet article, on consid ere des probl emes elliptiques d'ordre 2 sur des domaines tensoriels. Pour de tels domaines, on etudie la d eenition de bases d'ondelettes avec ordre d'approximation elev e, en mettant l'accent sur le choix des ondelettes pr es de la fronti ere pour optimiser le pr econditionnement diagonal des op erateurs elliptiques. AAn d'am eliorer les performances du pr econditionnement diagonal en base d'ondelettes, nous introduisons un pr econdition-nement presque diagonal obtenu en r esolvant des probl emes de Petrov-Galerkin locaux. L'eecacit e de ces m ethodes est mise en valeur en r esolvant, par une discr etisation uniforme, des probl emes elliptique d'ordre 2 a coeecients constants ou variables …