Anisotropic residual based a posteriori mesh adaptation in 2D: element based approach

TL;DR

This paper introduces an element-based anisotropic mesh adaptation method that effectively controls error norms without relying on a metric, demonstrating improved error management at increased computational cost.

Contribution

The paper presents a novel anisotropic a posteriori error estimator that adapts meshes through local modifications without using a metric, and introduces an $L^2$ variant with enhanced error control.

Findings

Favorable error control results compared to existing methods.

Increased CPU usage for better error management.

The $L^2$ estimator provides greater control over the $L^2$ error.

Abstract

An element based adaptation method is developed for an anisotropic a posteriori error estimator. The adaptation does not make use of a metric, but instead equidistributes the error over elements using local mesh modifications. Numerical results are reported, comparing with three popular anisotropic adaptation methods currently in use. It was found that the new method gives favourable results for controlling the energy norm of the error in terms of degrees of freedom at the cost of increased CPU usage. Additionally, we considered a new $L^2$ variant of the estimator. The estimator is shown to be conditionally equivalent to the exact $L^2$ error. We provide examples of adapted meshes with the $L^2$ estimator, and show that it gives greater control of the $L^2$ error compared with the original estimator.