We present and study techniques for investigating the spectra of linear differential operators on surfaces and flat domains using symmetric meshfree methods: meshfree methods that arise from finding norm-minimizing Hermite-Birkhoff interpolants in a Hilbert space. Meshfree methods are desirable for surface problems due to the increased difficulties associated with mesh creation and refinement on curved surfaces. While meshfree methods have been used for solving a wide range of partial differential equations (PDEs) in recent years, the spectra of operators discretized using radial basis functions (RBFs) often suffer from the presence of non-physical eigenvalues (spurious modes). This makes many RBF methods unhelpful for eigenvalue problems. We provide rigorously justified processes for finding eigenvalues based on results concerning the norm of the solution in its native space; specifically, only PDEs with solutions in the native space produce numerical solutions with bounded norms as the fill distance approaches zero. For certain problems, we prove that eigenvalue and eigenfunction estimates converge at a high-order rate. The technique we present is general enough to work for a wide variety of problems, including Steklov problems, where the eigenvalue parameter is in the boundary condition. Numerical experiments for a mix of standard and Steklov eigenproblems on surfaces with and without boundary, as well as flat domains, are presented, including a Steklov-Helmholtz problem.

翻译：本文提出并研究了一种利用对称网格无关方法探究曲面及平面域上线性微分算子谱的技术：该方法通过在希尔伯特空间中寻找范数最小化的Hermite-Birkhoff插值函数而构建。对于曲面问题，网格无关方法具有显著优势，因为在曲面上进行网格生成与细化存在较大困难。尽管近年来网格无关方法已广泛应用于各类偏微分方程求解，但基于径向基函数离散化的算子谱常受非物理特征值（伪模式）干扰，导致多数RBF方法难以有效处理特征值问题。我们基于解在其本征空间范数的理论结果，提出了严格论证的特征值求解流程：具体而言，仅当偏微分方程的解属于本征空间时，其数值解的范数才会在填充距离趋于零时保持有界。针对特定问题，我们证明了特征值与特征函数估计能以高阶速率收敛。所提方法具有足够普适性，可适用于包括Steklov问题在内的多种问题类型（其中特征参数出现于边界条件中）。本文展示了在含边界/无边界曲面及平面域上进行的标准与Steklov特征问题的数值实验，其中包含Steklov-Helmholtz问题的算例。