Adaptive refinement for eigenvalue problems based on an associated source problem

TL;DR

This paper presents an adaptive finite element method for efficiently approximating multiple eigenpairs of elliptic operators by solving a single source problem, offering a promising alternative to traditional eigenpair-based refinement.

Contribution

It introduces a novel adaptive refinement strategy driven by a source problem, reducing computational effort in eigenvalue computations for elliptic operators.

Findings

Effective in approximating large eigenpair collections

Reduces computational complexity compared to standard methods

Works well within hp-adaptive frameworks

Abstract

We introduce an adaptive finite element scheme for the efficient approximation of a (large) collection of eigenpairs of selfadjoint elliptic operators in which the adaptive refinement is driven by the solution of a single source problem -- the so-called landscape problem for the operator -- instead of refining based on the computed eigenpairs. Some theoretical justification for the approach is provided, and extensive empirical results indicate that it can provide an attractive alternative to standard adaptive schemes, particularly in the hp-adaptive environment.