# A new anisotropic mesh adaptation method based upon hierarchical a posteriori error estimates

## TL;DR

This paper introduces a novel anisotropic mesh adaptation method for finite element solutions of elliptic PDEs, utilizing hierarchical a posteriori error estimates to generate efficient meshes with minimal computational cost.

## Contribution

The paper presents a new anisotropic mesh adaptation strategy based on hierarchical a posteriori error estimates, improving efficiency and reliability over existing methods.

## Key findings

- Few Gauss-Seidel iterations suffice for accurate error approximation.
- The method outperforms strategies using local error estimators or recovered Hessians.
- Numerical examples demonstrate effectiveness in heat conduction problems with material jumps.

## Abstract

A new anisotropic mesh adaptation strategy for finite element solution of elliptic differential equations is presented. It generates anisotropic adaptive meshes as quasi-uniform ones in some metric space, with the metric tensor being computed based on hierarchical a posteriori error estimates. A global hierarchical error estimate is employed in this study to obtain reliable directional information of the solution. Instead of solving the global error problem exactly, which is costly in general, we solve it iteratively using the symmetric Gauss--Seidel (GS) method. Numerical results show that a few GS iterations are sufficient for obtaining a reasonably good approximation to the error for use in anisotropic mesh adaptation. The new method is compared with several strategies using local error estimators or recovered Hessians. Numerical results are presented for a selection of test examples and a mathematical model for heat conduction in a thermal battery with large orthotropic jumps in the material coefficients.