Inverse Iteration for the Laplace Eigenvalue Problem with Robin and Mixed Boundary Conditions

TL;DR

This paper applies inverse iteration to Laplace eigenvalue problems with Robin and mixed boundary conditions, proving convergence to principal eigenfunctions and eigenvalues, and introduces a related iterative method for an insulation model.

Contribution

It demonstrates convergence of inverse iteration for complex boundary conditions and proposes a new iterative approach for an insulation-related eigenvalue problem.

Findings

Convergence of iterates to principal eigenfunctions

Rayleigh quotients converge to principal eigenvalues

Proposed iterative method for insulation model

Abstract

We apply the method of inverse iteration to the Laplace eigenvalue problem with Robin and mixed Dirichlet-Neumann boundary conditions, respectively. For each problem, we prove convergence of the iterates to a non-trivial principal eigenfunction and show that the corresponding Rayleigh quotients converge to the principal eigenvalue. We also propose a related iterative method for an eigenvalue problem arising from a model for optimal insulation and provide some partial results.