A boundary integral algorithm for the Laplace Dirichlet–Neumann mixed eigenvalue problem

Abstract

We present a novel integral-equation algorithm for evaluation of Zaremba eigenvalues and eigenfunctions, that is, eigenvalues and eigenfunctions of the Laplace operator with mixed Dirichlet-Neumann boundary conditions; of course, (slight modifications of) our algorithms are also applicable to the pure Dirichlet and Neumann eigenproblems. Expressing the eigenfunctions by means of an ansatz based on the single layer boundary operator, the Zaremba eigenproblem is transformed into a nonlinear equation for the eigenvalue μ. For smooth domains the singular structure at Dirichlet-Neumann junctions is incorporated as part of our corresponding numerical algorithm-which otherwise relies on use of the cosine change of variables, trigonometric polynomials and, to avoid the Gibbs phenomenon that would arise from the solution singularities, the Fourier Continuation method (FC). The resulting numerical algorithm converges with high order accuracy without recourse to use of meshes finer than those resulting from the cosine transformation. For non-smooth (Lipschitz) domains, in turn, an alternative algorithm is presented which achieves highorder accuracy on the basis of graded meshes. In either case, smooth or Lipschitz boundary, eigenvalues are evaluated by searching for zero minimal singular values of a suitably stabilized discrete version of the single layer operator mentioned above. (The stabilization technique is used to enable robust non-local zero searches.) The resulting methods, which are fast and highly accurate for high-and low-frequencies alike, can solve extremely challenging two-dimensional Dirichlet, Neumann and Zaremba eigenproblems with high accuracies in short computing times-enabling, in particular, evaluation of thousands of eigenvalues and corresponding eigenfunctions for a given smooth or non-smooth geometry with nearly full double-precision accuracy.

Spectral structure of the Neumann–Poincaré operator on tori

“…As customary nowadays, some spectral analysis phenomena referring to the NP operator were first discovered via numerical experiments, see in particular [45,46]. From the abundant bibliography covering singular integral equations closely related to layer potentials we mention only a few titles: [101,3,23,26,44].…”

Optimization of Steklov-Neumann eigenvalues

“…The numerical implementation for the eigenvalues follows the boundaryintegral approach given for the mixed Steklov-Neumann problem in [2,3] and for the mixed Dirichlet-Neumann problem in [4] and is as follows.…”

High-Performance Computational Electromagnetics in Frequency-Domain and Time-Domain

“…Meaningful progress concerning well conditioned integral algorithms took place as a result of work sponsored by this contract, including rigorous mathematical theory and powerful numerical algorithms with applicability in a number of fields of science and engineering [12][13][14][15][16][17][18][19][20][21]. A variety of problems and configurations were thus considered, including problems of scattering by open surfaces [12][13][14]; improved integral methods for closed surfaces [15]; problems concerning propagation and scattering by penetrable scatterers [16]; studies of absorption properties of conducting materials containing asperities [17,18]; as well as new methods for evaluation of Laplace eigenfunctions on general domains and under challenging boundary conditions [19,20].…”

“…A variety of problems and configurations were thus considered, including problems of scattering by open surfaces [12][13][14]; improved integral methods for closed surfaces [15]; problems concerning propagation and scattering by penetrable scatterers [16]; studies of absorption properties of conducting materials containing asperities [17,18]; as well as new methods for evaluation of Laplace eigenfunctions on general domains and under challenging boundary conditions [19,20]. A rigorous convergence proof for the original methods [22] was provided in [21].…”

“…Consideration of various engineering and scientific configurations lead us to our recent development work concerning accurate numerical integral-equation methods for Laplace eigenvalue problems in general domains and under given boundary conditions [20]-with wide ranging applications in the fields of optics, photonics and antenna design, amongst many others, including analysis of waveguide and antenna problems. In particular this work provides a highly accurate eigensolver under the challenging Zaremba boundary conditions (for which no robust solvers existed before this work), and related Zaremba integral equation solvers [19] for the Helmholtz equation.…”