SCHWARZ METHODS WITH LOCAL REFINEMENT FOR THE P-VERSION FINITE-ELEMENT METHOD

In some applications, the accuracy of the numerical solution of an ell iptic problem needs to be increased only in certain parts of the domai n. In this paper, local refinement is introduced for an overlapping ad ditive Schwarz algorithm for the p-version finite element method. Both uniform and variable degree refinements are considered. The resulting algorithm is highly parallel and scalable. In two and three dimension s, we prove an optimal bound for the condition number of the iteration operator under certain hypotheses on the refinement region. This boun d is independent of the degree p, the number of subdomains N(r) and th e mesh size H. In the general two dimensional case, we prove an almost optimal bound with polylogarithmic growth in p.