Additive Schwarz Methods for the h-p Version of the Finite Element Method in Two Dimensions

Two additive Schwarz methods (ASMs) are proposed for the h-p version of the finite element method for two-dimensional elliptic problems in polygonal domains. One is based on generous overlapping of the h-version components (i.e., the linear nodal modes) and nonoverlapping of the p-version components (i.e., the high-order side modes and internal modes). Another is based on nonoverlapping for both the h-version and the p-version components. Their implementations are in parallel on the subdomain level for the h-version components and on the element level for 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 maxi(1 + ln (Hi pi/hi))2, where Hi is the diameter of the subdomain $\Omega_i$, hi is the characteristic diameter of the elements in $\Omega_i$, pi is the maximum polynomial degree used in $\Omega_i$, and p = maxi pi.