Reliable a-posteriori error estimators for $hp$-adaptive finite element approximations of eigenvalue/eigenvector problems

TL;DR

This paper develops reliable a-posteriori error estimators for $hp$-adaptive finite element methods applied to eigenvalue and eigenvector problems, extending previous $h$-adaptive techniques.

Contribution

It introduces a new framework for reliable and efficient a-posteriori error estimates in the $hp$-setting, building on boundary value problem analysis.

Findings

Provides error bounds for eigenvalue/eigenvector approximations

Extends $h$-adaptive estimators to $hp$-adaptive context

Combines boundary value problem analysis with eigenproblem estimation

Abstract

We present reliable a-posteriori error estimates for $hp$-adaptive finite element approximations of eigenvalue/eigenvector problems. Starting from our earlier work on $h$ adaptive finite element approximations we show a way to obtain reliable and efficient a-posteriori estimates in the $hp$-setting. At the core of our analysis is the reduction of the problem on the analysis of the associated boundary value problem. We start from the analysis of Wohlmuth and Melenk and combine this with our a-posteriori estimation framework to obtain eigenvalue/eigenvector approximation bounds.