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.

边界特征解的理论是针对边值问题而发展的。对于边值问题是通用的。Steklov-Poincaré运算符将一个域中拉普拉斯方程解的边界条件的值映射到另一个边界条件的值。特征值嵌入在Dirichlet到Neumann（DtN）映射中。DtN运算符称为Steklov运算符。在本文中，我们使用对偶边界元法/边界积分方程法（BEM / BIEM）研究了Steklov特征问题。首先，我们考虑一个循环域。为了分析得出上述形状的本征解，需要二维Laplace方程ln（r），通过极坐标系扩展为退化核。在针对Steklov特征问题的BIE的边界元素离散化之后，可以将其转换为标准线性特征方程。使用双BEM可以有效解决问题。最后，我们考虑环空。不仅考虑了Steklov问题，而且考虑了环形域的混合Steklov特征问题。