A correction adaptive two-grid finite element method for nonselfadjoint or indefinite elliptic problems

TL;DR

This paper introduces a correction adaptive two-grid finite element method for nonselfadjoint elliptic problems, improving accuracy and robustness over existing methods through a novel correction step.

Contribution

It presents a new correction step in the two-grid finite element method that enhances convergence and error reduction for challenging elliptic problems.

Findings

The corrected solution has higher-order L2-norm error compared to the energy-norm error.

The method converges with a contraction property for quasi-errors on successive meshes.

Numerical experiments show improved effectiveness and robustness over previous methods.

Abstract

We propose, analyze, and numerically validate a correction adaptive two-grid finite element method (CAT-GFEM) for nonselfadjoint or indefinite elliptic problems. In contrast to the adaptive two-grid finite element method (ATGFEM) of Li and Zhang [SIAM J. Sci. Comput., 43 (2021), pp. A908-A928], which is restricted to symmetric positive-definite problems, the proposed method introduces an additional correction step that solves a small-scale discrete residual problem on the coarse mesh. This step entails negligible additional computational cost and allows us to show that the L2-norm error of the corrected discrete solution is a higher-order of the energy-norm error of the discrete solution. Using this result, we prove a contraction property for a suitable sum of quasi-errors on two successive adaptive meshes and establish convergence of the method. Numerical experiments illustrate the improved effectiveness and robustness of our method in comparison with ATGFEM.