Numerical approaches to compute spectra of non-self adjoint operators in dimensions two and three

TL;DR

This paper develops and analyzes finite difference numerical methods to compute the spectra of non-self adjoint quadratic operators in two and three dimensions, addressing the challenges of instability and limited existing results.

Contribution

It introduces numerical approaches for multidimensional non-self adjoint operators and extends previous one-dimensional work to higher dimensions.

Findings

Finite difference methods effectively compute spectra in 2D and 3D.

Numerical results reveal spectral properties of non-self adjoint operators.

Analysis shows stability challenges and solution behaviors in higher dimensions.

Abstract

In this article we are interested for the numerical computation of spectra of non-self adjoint quadratic operators, in two and three spatial dimensions. Indeed, in the multidimensional case very few results are known on the location of the eignevalues. This leads to solve nonlinear eigenvalue problems. In introduction we begin with a review of theoretical results and numerical results obtained for the one dimensional case. Then we present the numerical methods developed to compute the spectra (finite difference discretization) for the two and three dimensional cases. The numerical results obtained are presented and analyzed. One difficulty here is that we have to compute eigenvalues of strongly non-self-adjoint operators which are unstable. This work is in continuity of a previous work in one spatial dimension.