Metric tensors for the interpolation error and its gradient in $L^p$ norm

TL;DR

This paper introduces a uniform method for deriving metric tensors in 2D to optimize interpolation errors and gradients in various $L^p$ norms, leading to optimal convergence in adaptive mesh generation.

Contribution

It presents a novel, unified approach to compute metric tensors for anisotropic adaptive meshes based on a posteriori error estimates in multiple $L^p$ norms.

Findings

Convergence rates are consistently optimal across tested scenarios.

The method effectively generates anisotropic meshes tailored to error estimates.

Numerical experiments validate the theoretical convergence properties.

Abstract

A uniform strategy to derive metric tensors in two spatial dimension for interpolation errors and their gradients in $L^p$ norm is presented. It generates anisotropic adaptive meshes as quasi-uniform ones in corresponding metric space, with the metric tensor being computed based on a posteriori error estimates in different norms. Numerical results show that the corresponding convergence rates are always optimal.