Expandable Local and Parallel Two-Grid Finite Element Scheme for the Stokes Equations

TL;DR

This paper introduces a new local and parallel two-grid finite element method for the Stokes equations, providing rigorous error estimates, an efficient iterative scheme, and numerical validation in 2D and 3D.

Contribution

It proposes a novel expandable two-grid finite element scheme with rigorous error analysis and an efficient iterative method for solving Stokes equations.

Findings

Achieves optimal convergence orders with specific iteration counts.

Provides error estimates for the proposed scheme.

Numerical tests confirm theoretical results in 2D and 3D.

Abstract

In this paper, we present a novel local and parallel two-grid finite element scheme for solving the Stokes equations, and rigorously establish its a priori error estimates. The scheme admits simultaneously small scales of subproblems and distances between subdomains and its expansions, and hence can be expandable. Based on the a priori error estimates, we provide a corresponding iterative scheme with suitable iteration number. The resulting iterative scheme can reach the optimal convergence orders within specific two-grid iterations ($O(|\ln H|^2)$ in 2-D and $O(|\ln H|)$ in 3-D) if the coarse mesh size $H$ and the fine mesh size $h$ are properly chosen. Finally, some numerical tests including 2-D and 3-D cases are carried out to verify our theoretical results.