Adaptive piecewise tensor product wavelets scheme for Laplace-interface problems

Abstract

A Laplace type boundary value problem is considered with a generally discontinuous diffusion coefficient. A domain decomposition technique is used to construct a piecewise tensor product wavelet basis that, when normalized w.r.t. the energy-norm, has Riesz constants that are bounded uniformly in the jumps. An adaptive wavelet Galerkin method is applied to solve the boundary value problem with the best nonlinear approximation rate from the basis, in linear computational complexity. Although the solutions are far from smooth, numerical experiments in two dimensions show rates as for a one-dimensional smooth solution, the latter being possible because of the tensor product construction.