Numerical approaches to compute spectra of non-self adjoint operators in dimensions two and three
Fatima Aboud, François Jauberteau, Didier Robert
TL;DR
The paper tackles the numerical computation of spectra for non-self-adjoint quadratic operator families in two and three spatial dimensions. It develops finite-difference discretizations for both homogeneous Dirichlet and periodic boundary conditions, reducing nonlinear eigenproblems to linearized block forms involving sparse matrices $H_{0}$ and $H_{1}$ and solving with LAPACK. Key findings show that eigenvalues tend to lie in sectors consistent with one-dimensional theory, with the real parts increasing as the coupling parameter $c$ grows and the imaginary spread increasing with grid resolution or domain size; periodic cases exhibit symmetry patterns depending on coefficients. The work extends spectral insights to higher dimensions and demonstrates pseudospectral analysis as a practical tool for unstable spectra, offering computational avenues for studying spectra of non-self-adjoint quadratic operators 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.