Domain decomposition preconditioning for elliptic problems with jumps in coefficients

We propose an effective iterative preconditioning method to solve elliptic problems with jumps in coefficients. The algorithm is based on the additive Schwarz method (ASM). First, we consider a domain decomposition method without cross points on the interfaces between subdomains and next, we treat the cross points case. In both cases the main computational cost is an implementation of preconditioners for the Laplace operator in whole domain and in subdomains. Iterative convergence is independent of jumps in coefficients and mesh size. Several numerical examples are given to demonstrate the performance of proposed algorithm and theory developed in the paper.