Verified eigenvalue evaluation for Laplacian over polygonal domain of arbitrary shape

TL;DR

This paper introduces a novel FEM-based algorithm that provides guaranteed bounds for Laplacian eigenvalues on arbitrary polygonal domains, including those with singularities, improving accuracy over classical methods.

Contribution

The paper develops a new computable approach for eigenvalue bounds on arbitrary shapes, incorporating a priori error estimates and interval arithmetic for guaranteed results.

Findings

Effective eigenvalue bounds for non-convex and singular domains

Demonstrated accuracy with examples like L-shaped and crack domains

Enhanced eigenvalue estimation precision using Lehmann's theorem

Abstract

The finite element method(FEM) is applied to bound leading eigenvalues of Laplace operator over polygonal domain. Compared with classical numerical methods, most of which can only give concrete eigenvalue bounds over special domain of symmetry, our proposed algorithm can provide concrete eigenvalue bounds for domain of arbitrary shape, even in the case that eigenfunction has singularity. The problem of eigenvalue estimation is solved in two steps. First, we construct computable a priori error estimation for FEM solution of Poisson's problem even for non-convex domain with re-entrant corner. Second, a new computable lower and upper bounds is developed for eigenvalues. As the interval arithmetic is implemented in the FEM computation, the desired eigenvalue bounds can be expected to be mathematically correct. The Lehmann's theorem is also adopted to sharpen the eigenvalue bounds with high precision. At the end of this paper, we illustrate several computation examples, such as the case of L-shaped domain and crack domain, to demonstrate the efficiency and flexibility of proposed method.