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

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.