In this study, we propose a novel approach based on the dual reciprocity boundary element method (DRBEM) to approximate the solutions of various Steklov eigenvalue problems. The method consists in weighting the governing differential equation with the fundamental solutions of the Laplace equation where the definition of interior nodes is not necessary for the solution on the boundary. DRBEM constitutes a promising tool to characterize such problems due to the fact that the boundary conditions on part or all of the boundary of the given flow domain depend on the spectral parameter. The matrices resulting from the discretization are partitioned in a novel way to relate the eigenfunction with its flux on the boundary where the spectral parameter resides. The discretization is carried out with the use of constant boundary elements resulting in a generalized eigenvalue problem of moderate size that can be solved at a smaller expense compared to full domain discretization alternatives. We systematically investigate the convergence of the method by several experiments including cases with selfadjoint and non-selfadjoint operators. We present numerical results which demonstrate that the proposed approach is able to efficiently approximate the solutions of various mixed Steklov eigenvalue problems defined on arbitrary domains.

在这项研究中，我们提出了一种基于对等互惠边界元方法（DRBEM）的新颖方法，以逼近各种Steklov特征值问题的解决方案。该方法包括用拉普拉斯方程的基本解对控制微分方程加权，其中内部节点的定义对于边界解不是必需的。由于给定流域的部分或全部边界上的边界条件取决于光谱参数，因此DRBEM成为表征此类问题的有前途的工具。由离散化产生的矩阵以新颖的方式进行划分，以将特征函数与其通量关联到光谱参数所驻留的边界上。离散化是通过使用恒定边界元素进行的，从而导致中等大小的广义特征值问题，与全域离散化替代方案相比，可以用较小的费用解决该问题。我们通过几个实验，包括自伴算子和非自伴算子的案例，系统地研究了该方法的收敛性。我们提供的数值结果表明，所提出的方法能够有效地逼近定义在任意域上的各种混合Steklov特征值问题的解决方案。