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

TL;DR

This paper introduces a mixed basis perturbation method to efficiently approximate the spectrum of the Laplace operator, especially for complex geometries with Neumann boundary conditions, by reducing the problem to surface calculations.

Contribution

It proposes a novel mixed basis approach that treats boundary effects as perturbations, enabling simultaneous computation of multiple eigenvalues with reduced complexity.

Findings

Effective approximation of Laplace eigenvalues for 2D geometries

Reduces computational complexity by separating boundary from volume

Applicable to problems with Neumann boundary conditions

Abstract

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.