Two-grid algorithms for singularly perturbed reaction-diffusion problems on layer adapted meshes

TL;DR

This paper introduces a two-grid finite element method for singularly perturbed reaction-diffusion problems, achieving uniform convergence and efficiency by combining quasilinearization with layer-adapted meshes.

Contribution

It develops a novel two-grid approach that combines Bellman-Kalaba quasilinearization with Axelsson-Xu finite element methods on layer-adapted meshes, ensuring uniform convergence.

Findings

The method is uniformly convergent across different layer-adapted meshes.

The two-grid approach achieves the same accuracy as solving the nonlinear problem directly on fine meshes.

Numerical results confirm theoretical convergence properties.

Abstract

We propose a new two-grid approach based on Bellman-Kalaba quasilinearization and Axelsson-Xu finite element two-grid method for the solution of singularly perturbed reaction-diffusion equations. The algorithms involve solving one inexpensive problem on coarse grid and solving on fine grid one linear problem obtained by quasilinearization of the differential equation about an interpolant of the computed solution on the coarse grid. Different meshes (of Bakhvalov, Shishkin and Vulanovi\'c types) are examined. All the schemes are uniformly convergent with respect to the small parameter. We show theoretically and numerically that the global error of the two-grid method is the same as of the nonlinear problem solved directly on the fine layer-adapted mesh.