The spectrum of the Steklov-Helmholtz operator

Abstract

We present a wavenumber-robust strategy for computing Steklov eigenpairs of the Helmholtz operator $-Δ-μ^2$. As the wavenumber $μ\rightarrow μ_D$ from below (where $μ_D^2 $ is a Dirichlet- Laplace eigenvalue of multiplicity $\ell$), the lowest $\ell$ Steklov-Helmholtz eigenvalues diverge to $-\infty$. Computationally, the Steklov-Helmholtz eigenvalue problem becomes severely ill-conditioned when $μ\approx μ_D$. We first reformulate the problem in terms of a suitably-defined Dirichlet-to-Neumann map. We then use an indirect approach based on a single layer ansatz. The discrete single layer matrix is nearly singular close to exceptional wavenumbers, and we use a reduced singular value decomposition to avoid the consequent ill-conditioning. For smooth domains, convergence of our eigenvalue solver is spectral. We use this method (called the BIO-MOD approach) for shape optimization of scale-invariant Steklov-Helmholtz problems and prove that the disk maximizes the second eigenvalue under appropriate scaling. For curvilinear polygons, we use polynomially-graded meshes rather than uniform meshes. As a proof of concept, we also implemented BIO-MOD using RCIP quadratures (using the ChunkIE implementation). The BIO-MOD approach successfully removes ill-conditioning near exceptional wavenumbers, and very high eigenvalue accuracy (up to 10 digits for polygons, arbitrary precision accuracy for smooth domains) is observed. We deploy our approach to computationally study the spectral geometry of the Steklov-Helmholtz operator, including some questions about spectral asymptotics and spectral optimization.