Convergence rates for adaptive finite elements

TL;DR

This paper proves that using newest-vertex bisection, adaptive finite element meshes can effectively equidistribute error for functions with regular and singular parts, leading to quasi-optimal convergence rates for AFEM.

Contribution

It introduces a method to construct error-equidistributing meshes for functions with singularities, ensuring quasi-optimal convergence in adaptive finite element methods.

Findings

Meshes equidistribute error in $H^1$-norm for functions with singularities.

Convergence rates for AFEM with Lagrange elements are established.

Meshes are quasi-optimal for the class of functions considered.

Abstract

In this article we prove that it is possible to construct, using newest-vertex bisection, meshes that equidistribute the error in $H^1$-norm, whenever the function to approximate can be decomposed as a sum of a regular part plus a singular part with singularities around a finite number of points. This decomposition is usual in regularity results of Partial Differential Equations (PDE). As a consequence, the meshes turn out to be quasi-optimal, and convergence rates for adaptive finite element methods (AFEM) using Lagrange finite elements of any polynomial degree are obtained.