Fourier Method for Approximating Eigenvalues of Indefinite Stekloff Operator

TL;DR

This paper presents a Fourier-based numerical method leveraging FFT to efficiently approximate Stekloff eigenvalues linked to the Helmholtz equation, reducing computational complexity and improving accuracy.

Contribution

The paper introduces a novel FFT-based approach for computing Stekloff eigenvalues, simplifying the process and enhancing efficiency compared to traditional methods.

Findings

Efficient computation of Stekloff eigenvalues using FFT.

Reduction in computational cost for Helmholtz eigenvalue problems.

Clear and practical implementation of the Fourier method.

Abstract

We introduce an efficient method for computing the Stekloff eigenvalues associated with the Helmholtz equation. In general, this eigenvalue problem requires solving the Helmholtz equation with Dirichlet and/or Neumann boundary condition repeatedly. We propose solving the related constant coefficient Helmholtz equation with Fast Fourier Transform (FFT) based on carefully designed extensions and restrictions of the equation. The proposed Fourier method, combined with proper eigensolver, results in an efficient and clear approach for computing the Stekloff eigenvalues.