Outlier-free isogeometric discretizations for Laplace eigenvalue problems: closed-form eigenvalue and eigenvector expressions

TL;DR

This paper derives explicit formulas for eigenvalues and eigenvectors of isogeometric discretizations of the Laplace operator, revealing their matrix structures and confirming the outlier-free nature of certain spline spaces.

Contribution

It provides closed-form eigenvalue and eigenvector expressions for outlier-free isogeometric discretizations, and characterizes the matrix structures involved.

Findings

Eigenvalues are explicit samples of spectral symbols.

Matrix structures are Toeplitz-minus-Hankel or Toeplitz-plus-Hankel.

Optimal spline spaces are confirmed to be outlier-free.

Abstract

We derive explicit closed-form expressions for the eigenvalues and eigenvectors of the matrices resulting from isogeometric Galerkin discretizations based on outlier-free spline subspaces for the Laplace operator, under different types of homogeneous boundary conditions on bounded intervals. For optimal spline subspaces and specific reduced spline spaces, represented in terms of B-spline-like bases, we show that the corresponding mass and stiffness matrices exhibit a Toeplitz-minus-Hankel or Toeplitz-plus-Hankel structure. Such matrix structure holds for any degree p and implies that the eigenvalues are an explicitly known sampling of the spectral symbol of the Toeplitz part. Moreover, by employing tensor-product arguments, we extend the closed-form property of the eigenvalues and eigenvectors to a d-dimensional box. As a side result, we have an algebraic confirmation that the considered optimal and reduced spline spaces are indeed outlier-free.