An Expandable Local and Parallel Two-Grid Finite Element Scheme

TL;DR

This paper introduces a scalable, parallel two-grid finite element scheme for elliptic problems, demonstrating optimal convergence and efficiency on large parallel systems through theoretical analysis and numerical validation.

Contribution

It presents a novel expandable local and parallel two-grid finite element scheme based on superposition, suitable for large parallel computing environments, with proven optimal convergence rates.

Findings

Achieves optimal convergence within logarithmic factors of mesh size

Suitable for implementation on large parallel computer systems

Numerical results confirm theoretical error estimates

Abstract

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.