Analytical solutions to some generalized and polynomial eigenvalue problems

TL;DR

This paper derives analytical solutions for generalized and polynomial eigenvalue problems arising from finite element and isogeometric analysis, extending classical results and improving understanding of eigenstructure in these contexts.

Contribution

It generalizes analytical eigenpair solutions to GEVPs from FEM and IGA matrices, and extends eigenvector-eigenvalue identities to these problems.

Findings

Analytical solutions for GEVPs from FEM and IGA matrices.

Eigenvector-eigenvalue identities extended to GEVPs.

Better numerical methods for IGA matrices suggested.

Abstract

It is well-known that the finite difference discretization of the Laplacian eigenvalue problem $-\Delta u = \lambda u$ leads to a matrix eigenvalue problem (EVP) $A x= \lambda x$ where the matrix $A$ is Toeplitz-plus-Hankel. Analytical solutions to tridiagonal matrices with various boundary conditions are given in Strang and MacNamara \cite{strang2014functions}. We generalize the results and develop analytical solutions to the generalized matrix eigenvalue problems (GEVPs) $A x= \lambda Bx$ which arise from the finite element method (FEM) and isogeometric analysis (IGA). The FEM matrices are corner-overlapped block-diagonal while the IGA matrices are almost Toeplitz-plus-Hankel. In fact, IGA with a correction that results in Toeplitz-plus-Hankel matrices gives a better numerical method. In this paper, we focus on finding the analytical eigenpairs to the GEVPs while developing better numerical methods is our motivation. Analytical solutions are also obtained for some polynomial eigenvalue problems (PEVPs). Lastly, we generalize the eigenvector-eigenvalue identity (rediscovered and coined recently for EVPs) for GEVPs and derive some trigonometric identities.