Numerical determination of the fundamental eigenvalue for the Laplace operator on a spherical domain

Abstract

Methods for obtaining approximate solutions for the fundamental eigenvalue of the Laplace-Beltrami operator (i.e., the membrane eigenvalue problem for the vibration equation) on the unit spherical surface are developed. Two types of spherical surface domains are considered: (1) the interior of a spherical triangle, and (2) the exterior of a great circle arc extending for less thanπ radians (a spherical surface with a slit). In both cases, zero boundary conditions are imposed. In order to solve the resulting second-order elliptic partial differential equations in two independent variables, a finite difference approximation is employed. The fundamental eigenvalue is approximated by iteration utilizing the power method and point successive overrelaxation. Some numerical results are given and compared, in certain special cases, with analytical solutions to the eigenvalue problem. The significance of the numerical eigenvalue results is discussed in terms of the singularities in the solution of three-dimensional boundary-value problems near a polyhedral corner of the domain.