ADDITIVE SCHWARZ METHODS FOR THE H-P VERSION OF THE FINITE-ELEMENT METHOD IN 2 DIMENSIONS

Two additive Schwarz methods (ASMs) are proposed for the h-p version o f the finite element method for two-dimensional elliptic problems in p olygonal domains. One is based on generous overlapping of the h-versio n components (i.e., the linear nodal modes) and nonoverlapping of the p-version components (i.e., the high-order side modes and internal mod es). Another is based on nonoverlapping for both the h-version and the p-version components. Their implementations are in parallel on the su bdomain level for the h-version components and on the element level fo r the p-version components. The condition number for the first method is of order O(1 + ln p)(2), and for the second one is max(i)(1 + ln(H( i)p(i)/h(i)))(2), where H-i is the diameter of the subdomain Omega(i), h(i) is the characteristic diameter of the elements in Omega(i), p(i) is the maximum polynomial degree used in Omega(i), and p = max(i)p(i) .