A Mixed Basis Perturbation Approach to Approximate the Spectrum of Laplace Operator

This paper presents a mixed basis approach for Laplace eigenvalue problems, which treats the boundary as a perturbation of the free Laplace operator. The method separates the boundary from the volume via a generic function that can be pre-calculated and thereby effectively reduces the complexity of the problem to a calculation over the surface. Several eigenvalues are retrieved simultaneously. The method is applied to several 2 dimensional geometries with Neumann boundary conditions. NOMENCLATURE D Diffusion constant. ∆ Laplace operator/Laplace matrix. λ Eigenvalue of ∆ ∗Address all correspondence to this author. ∆F Free Laplace operator. S Surface operator. Φ(x) Fundamental solution of the Laplace equation. σi(x) Surface charge distribution. Φi(x) Solution to the charge distribution σi. Ψi Eigenfunction to ∆F . INTRODUCTION In physics and chemistry many situations with non-trivial boundary conditions give rise to a Laplace eigenvalue problem that is not analytically solvable. In such situations numerical methods, such as finite element methods (FEM) [1], fast multi-pole methods (FMM) [2], analytic element methods (AEM) [3], boundary element methods (BEM) [4], and boundary approximation methods (BAM) [5] (among many others) are used. In this paper an alternative method is presented that