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

Wavelet methods allow us to combine high-order accuracy, efficient preconditioning techniques, and adaptive approximations in order to solve efficiently elliptic operator equations. Many difficulties remain, in particular, related to the adaptation of wavelet decompositions to bounded domains with prescribed boundary conditions, leading to possibly high constants in the ${\cal O}(1)$ preconditioning. In this paper we consider the framework of conforming domain decomposition to generate our wavelet bases and second-order operators. We emphasize the choice of the wavelets near the boundary of the tensor product reference domain in order to optimize the efficiency of the diagonal preconditioning of elliptic operators. In order to improve the constants obtained by such diagonal preconditionings, we propose to take into account interactions between the scales through the computation of a sparse approximate inverse (SPAI) on a set of nonzero entries obtained from the compression of the operator itself in the wavelet basis. The efficiency of these methods is illustrated by solving elliptic second-order problems with variable or constant coefficients and homogeneous boundary conditions on a uniform discretization. Finally, we propose a coupling of the iterative solver with an adaptive space refinement technique. On the Laplacian model problem, our experiments show that this algorithm generates an optimal nonlinear approximation of the solution.