An adaptive tensor product wavelet method for solving second order elliptic problems with jump coefficients

We consider a Laplace type boundary value problem with a generally discontinuous diffusion coefficient on a domain that is a non-overlapping domain decomposition. Each subdomain is a hypercube and we equip the subdomain with tensor product wavelet basis. By the application of extension operator, we conctruct a basis on a domain from tensor product wavelet basis on subdomains. Adaptive piecewise tensor product wavelet scheme is applied for solving Poisson’s equation with jump coefficient on the subdomains. It will be demonstrated that the resulting approximations converge in the adapted energy norm with the best nonlinear approximation rate from the span of the best piecewise tensor product wavelets, in linear computational complexity.