An Expandable Local and Parallel Two-Grid Finite Element Scheme
This work addresses the need for scalable parallel finite element methods for elliptic problems, offering an incremental improvement over existing local and parallel schemes.
The paper proposes an expandable local and parallel two-grid finite element scheme for elliptic problems that is easily implementable on large parallel systems. The scheme achieves optimal convergence orders within |ln H|^2 or |ln H| two-grid iterations in 2-D or 3-D, respectively, with proper mesh sizes.
An expandable local and parallel two-grid finite element scheme based on superposition principle for elliptic problems is proposed and analyzed in this paper by taking example of Poisson equation. Compared with the usual local and parallel finite element schemes, the scheme proposed in this paper can be easily implemented in a large parallel computer system that has a lot of CPUs. Convergence results base on $H^1$ and $L^2$ a priori error estimation of the scheme are obtained, which show that the scheme can reach the optimal convergence orders within $|\ln H|^2$ or $|\ln H|$ two-grid iterations if the coarse mesh size $H$ and the fine mesh size $h$ are properly configured in 2-D or 3-D case, respectively. Some numerical results are presented at the end of the paper to support our analysis.