[Paper Review] Spectral Indicator Method for A Non-selfadjoint Steklov Eigenvalue Problem

Juan Liu, Jiguang Sun+1 more|arXiv (Cornell University)|Apr 7, 2018

Advanced Numerical Methods in Computational Mathematics24 references3 citations

TL;DR

This paper proposes a finite element method combined with a modified spectral indicator method (RIM) to compute complex eigenvalues for a non-selfadjoint Steklov eigenvalue problem arising in inverse scattering theory. By reducing the problem to a smaller boundary-only discrete system, the method efficiently computes eigenvalues using spectral projection via contour integrals, achieving second-order convergence and validating results on circular, square, and L-shaped domains.

ABSTRACT

We propose an efficient numerical method for a non-selfadjoint Steklov eigenvalue problem. The Lagrange finite element is used for discretization. The convergence is proved using the spectral perturbation theory for compact operators. The non-sefadjointness of the problem leads to non-Hermitian matrix eigenvalue problem. Due to the existence of complex eigenvalues and lack of a priori spectral information, we propose a modified version of the recently developed spectral indicator method to compute (complex) eigenvalues in a given region on the complex plane. In particular, to reduce computational cost, the problem is transformed into a much smaller matrix eigenvalue problem involving the unknowns only on the boundary of the domain. Numerical examples are presented to validate the effectiveness of the proposed method.

Motivation & Objective

• To develop a finite element method for a non-selfadjoint Steklov eigenvalue problem with complex spectral parameters.
• To address the challenge of computing complex eigenvalues without prior spectral information in non-Hermitian matrix eigenvalue problems.
• To reduce computational cost by transforming the problem into a smaller matrix eigenvalue problem involving only boundary unknowns.
• To extend the recursive integral method (RIM) for non-selfadjoint problems and prove convergence using spectral perturbation theory for compact operators.
• To provide the first comprehensive numerical study of a non-selfadjoint Steklov eigenvalue problem with both theoretical analysis and computational validation.

Proposed method

• A linear Lagrange finite element method is applied to the variational formulation of the non-selfadjoint Steklov eigenvalue problem.
• The problem is transformed into a non-Hermitian generalized eigenvalue problem through Galerkin discretization.
• A boundary-only reduction is derived, resulting in a smaller matrix eigenvalue problem that depends only on unknowns on ∂Ω.
• The recursive integral method (RIM) is adapted to compute complex eigenvalues in a specified region of the complex plane via Cauchy contour integrals.
• Spectral projection is used to define an indicator function that determines whether a region contains eigenvalues, enabling adaptive subdivision.
• The method uses a stopping criterion based on region size (e.g., d₀ = 1e−9) to refine eigenvalue approximations.

Experimental results

Research questions

• RQ1 Can a finite element method be rigorously proven to converge for a non-selfadjoint Steklov eigenvalue problem?
• RQ2 How can complex eigenvalues be efficiently computed when no a priori spectral information is available?
• RQ3 Can the computational cost be significantly reduced by restricting the discrete problem to the boundary only?
• RQ4 What is the convergence rate of the finite element approximation for non-selfadjoint Steklov eigenvalues with varying domain geometries?
• RQ5 How does the modified RIM perform in computing complex eigenvalues for non-Hermitian problems arising from Helmholtz-type equations?

Key findings

• Second-order convergence is achieved for the unit circle and square domains when solving the non-selfadjoint Steklov eigenvalue problem with n(x) = 4 + 4i.
• The second eigenvalue of the L-shaped domain shows a reduced convergence rate, indicating lower regularity of the associated eigenfunction.
• For the unit circle with n(x) = 4 + 4i, the first eigenvalue converges to approximately 0.6866 + 2.4953i as h → 0.
• For the square domain with n(x) = 4 + 4i, the first eigenvalue converges to approximately 0.5144 + 2.8824i as h → 0.
• The computed eigenvalues for the L-shaped domain stabilize to within 1e−4 in the real and imaginary parts for h ≤ 0.0149.
• The method successfully computes multiple complex eigenvalues with high accuracy, and the boundary-only formulation significantly reduces computational cost.