An Additive Schwarz Method for the h-p Version of the Finite Element Method in Three Dimensions

Abstract

In this paper, we study the additive Schwarz method for the h-p version of the finite element method in three dimensions. The main idea is to treat separately the h-version (linear) components and the p-version (high-order) components by a vertex-based method. It can also be viewed as a three-level method with the level being the linear finite element approximation on the coarse mesh, the linear finite element approximation on the fine mesh, and the high-order finite element approximation on the fine mesh, respectively. The resulting algorithm can be implemented in parallel on the subdomain level for the h-version components and on the element level for the p-version components. The condition number is of order $\max\limits_i(1 + ln Hipi / hi )2, where Hi stands for the diameter of the subdomain $\Omega_i$, hi is the diameter of the elements in $\Omega_i$, and pi is the maximum of the polynomial degrees used in $\Omega_i$.