Finite element method to solve the spectral problem for arbitrary self-adjoint extensions of the Laplace-Beltrami operator on manifolds with a boundary

TL;DR

This paper introduces a numerical scheme for computing the spectra of various self-adjoint extensions of the Laplace-Beltrami operator on manifolds with boundary, applicable in any dimension and accommodating diverse boundary conditions.

Contribution

It presents a unified algorithm based on quadratic forms for a broad class of self-adjoint extensions, with proven convergence and effective implementation in 2D.

Findings

The scheme converges reliably for different boundary conditions.

Numerical examples demonstrate the algorithm's versatility and effectiveness.

The method applies to manifolds of any dimension.

Abstract

A numerical scheme to compute the spectrum of a large class of self-adjoint extensions of the Laplace-Beltrami operator on manifolds with boundary in any dimension is presented. The algorithm is based on the characterisation of a large class of self-adjoint extensions of Laplace-Beltrami operators in terms of their associated quadratic forms. The convergence of the scheme is proved. A two-dimensional version of the algorithm is implemented effectively and several numerical examples are computed showing that the algorithm treats in a unified way a wide variety of boundary conditions.