# Benchmark computation of eigenvalues with large defect for non-selfadjoint elliptic differential operators

## TL;DR

This paper introduces benchmark problems for non-selfadjoint elliptic eigenvalue problems with large defect, analyzing the convergence of finite element methods and verifying theoretical estimates through numerical experiments.

## Contribution

It provides a set of benchmark problems with discontinuous coefficients and mixed boundary conditions, and investigates the convergence behavior of finite element methods for these problems.

## Key findings

- Finite element methods converge as expected with respect to discretization parameters.
- The ascent of eigenvalues significantly affects the eigenvalue approximation.
- Considering the mean value improves the accuracy of eigenvalue estimates.

## Abstract

In this paper we present benchmark problems for non-selfadjoint elliptic eigenvalue problems with large defect and ascent. We describe the derivation of the benchmark problem with a discontinuous coefficient and mixed boundary conditions. Numerical experiments are performed to investigate the convergence of a Galerkin finite element method with respect to the discretization parameters, the regularity of the problem, and the ascent of the eigenvalue. This allows us to verify the sharpness of the theoretical estimates from the literature with respect to these parameters. We provide numerical evidence about the size of the ascent and show that it is important to consider the mean value for the eigenvalue approximation.