A boundary integral equation method for Steklov eigenvalue problems for smooth planar domains

Abstract: In this paper, we study the computational question of whether the Steklov spectrum of smooth simply connected planar domains can be approximated accurately by a boundary-only formulation based on harmonic conjugation. For the unit disk, the Dirichlet-to-Neumann operator can be written explicitly in terms of the classical conjugation operator. We show how this viewpoint extends to general bounded and unbounded simply connected domains through the generalized conjugation operator defined through the boundary integral equation with the generalized Neumann kernel. Combined with Fourier differentiation on an equidistant boundary grid, this leads to a dense algebraic eigenvalue problem for the boundary traces of Steklov eigenfunctions. The resulting method uses only boundary data, treats interior and exterior problems in a unified way, and reconstructs eigenfunctions in the domain by harmonic extension. Numerical experiments on benchmark domains and on parameter-dependent smooth families, including ellipses and star-like curves, show high accuracy for smooth boundaries and illustrate how the Steklov spectrum changes with geometry.