Finite element method to solve the spectral problem for arbitrary self-adjoint extensions of the Laplace-Beltrami operator on manifolds with a boundary
This work provides a unified numerical method for spectral problems on manifolds, relevant to researchers in geometric analysis and numerical PDEs.
The paper presents a numerical scheme for computing the spectrum of self-adjoint extensions of the Laplace-Beltrami operator on manifolds with boundary, proving convergence and demonstrating effectiveness on 2D examples with various boundary conditions.
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.