New local and parallel finite element algorithm based on the partition of unity

Abstract

In this study, based on a combination of the two-grid method and the partition of unity-based domain decomposition method, we propose a new local and parallel finite element algorithm for the elliptic boundary value problem. The proposed method has three key features: (1) it inherits the flexibility and controllability of domain decomposition based on the partition of unity; (2) global fine grid correction is replaced by solving a series of locally defined approximate residual problems with homogeneous Dirichlet boundary conditions on some finer grids; (3) a global continuous finite element solution is constructed by solving a coarse grid correction problem and by assembling all the local solutions together using the partition of unity subordinate. Under appropriate assumptions, the optimal error estimates in L2 and the energy norms are proved by new analytical results. In addition, several numerical simulations are presented to demonstrate the high efficiency and flexibility of the new algorithm.