Eigenvalues and eigenfunctions of the Laplacian via inverse iteration with shift

TL;DR

This paper introduces a novel iterative method inspired by inverse iteration with shift to compute eigenvalues and eigenfunctions of the Laplacian on arbitrary domains, offering a direct functional analysis approach that is efficient and avoids finite-dimensional approximations.

Contribution

The paper presents a new functional analysis-based iterative method for Laplacian eigenvalue problems that bypasses finite linear algebra approximations and introduces an efficient alternative Rayleigh quotient expression.

Findings

Method converges uniformly away from nodal surfaces.

Produces eigenvalues and eigenfunctions with minimal computational effort.

Enables spectral decomposition of functions in L^2(Ω).

Abstract

In this paper we present an iterative method, inspired by the inverse iteration with shift technique of finite linear algebra, designed to find the eigenvalues and eigenfunctions of the Laplacian with homogeneous Dirichlet boundary condition for arbitrary bounded domains $\Omega\subset R^{N}$. This method, which has a direct functional analysis approach, does not approximate the eigenvalues of the Laplacian as those of a finite linear operator. It is based on the uniform convergence away from nodal surfaces and can produce a simple and fast algorithm for computing the eigenvalues with minimal computational requirements, instead of using the ubiquitous Rayleigh quotient of finite linear algebra. Also, an alternative expression for the Rayleigh quotient in the associated infinite dimensional Sobolev space which avoids the integration of gradients is introduced and shown to be more efficient. The method can also be used in order to produce the spectral decomposition of any given function $u\in L^{2}(\Omega)$.