Analytical and numerical studies for solving Steklov eigenproblems by using the boundary integral equation method/boundary element method

Abstract

The theory of boundary eigensolutions is developed for boundary value problems. It is general for boundary value problem. Steklov-Poincaré operator maps the values of a boundary condition of the solution of the Laplace equation in a domain to the values of another boundary condition. The eigenvalue is imbedded in the Dirichlet to Neumann (DtN) map. The DtN operator is called the Steklov operator. In this paper, we study the Steklov eigenproblems by using the dual boundary element method/boundary integral equation method (BEM/BIEM). First, we consider a circular domain. To analytically derive the eigensolution of the above shape, the closed-form fundamental solution of the 2D Laplace equation, ln(r), is expanded into degenerate kernel by using the polar coordinate system. After the boundary element discretization of the BIE for the Steklov eigenproblem, it can be transformed to a standard linear eigenequation. Problems can be effectively solved by using the dual BEM. Finally, we consider the annulus. Not only the Steklov problem but also the mixed Steklov eigenproblem for an annular domain has been considered.