Eigenvalues and eigenfunctions of the Laplacian via inverse iteration with shift

Details Eigenvalues and eigenfunctions of the Laplacian via inverse iteration with shift Eigenvalues and eigenfunctions of the Laplacian via inverse iteration with shift

2010-11-14

arxiv.org/abs/1011.3266v1

Mathematics

Spectral Theory

Scientific paper

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 $\Omega \subset\mathbb{R}^{N}$. Uniform convergence away from nodal surfaces is obtained and used in order to produce a faster and more accurate algorithm for computing the eigenvalues with minimal computational requirements, instead of using the ubiquitous Rayleigh quotient in finite linear algebra. The method can also be used in order to produce the spectral decomposition of any given function $u\in L^2(\Omega)$.

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