Local and Parallel Finite Element Algorithms Based on the Partition of Unity for the Stokes Problem

Abstract

By combining the techniques of the two-grid method and the partition of unity, we derive two local and parallel finite element algorithms for the Stokes problem. The most interesting features of these algorithms are (1) the partition of unity technique introduces a framework for domain decomposition; (2) only a series of local residual problems need to be solved on these subdomains in parallel, meanwhile requiring very little communication; (3) a globally continuous finite element solution is constructed by combining all the local solutions via the partition of unity functions. The optimal error estimates in $L^2$ and energy norms are proved under some assumptions. Also, several numerical simulations are presented to demonstrate the effectiveness and flexibility of the new algorithms.