In this paper, we consider the numerical approximation of the Steklov eigenvalue problem that arises in inverse acoustic scattering. The underlying scattering problem is for an inhomogeneous isotropic medium. These eigenvalues have been proposed to be used as a target signature since they can be recovered from the scattering data. A Galerkin method is studied where the basis functions are the Neumann eigenfunctions of the Laplacian. Error estimates for the eigenvalues and eigenfunctions are proven by appealing to Weyl's Law. We will test this method against separation of variables in order to validate the theoretical convergence. We also consider the inverse spectral problem of estimating/recovering the refractive index from the knowledge of the Steklov eigenvalues. Since the eigenvalues are monotone with respect to a real-valued refractive index implies that they can be used for non-destructive testing. Some numerical examples are provided for the inverse spectral problem.
翻译:在本文中, 我们考虑在反声学散射中产生的Steklov egenvaly问题的数字近似值。 潜在的散射问题在于一个不相异的异热带介质。 这些egenvaly建议作为目标标志使用, 因为可以从散射数据中回收它们。 在基函数是Laplacian 的Neumann 元元功能的情况下, 研究了一种 Galerkin 方法。 向Weyl 法律上诉可以证明对egenvaly和egenwids的错误估计值。 我们将测试这种方法, 防止变量分离, 以验证理论趋同。 我们还考虑了从 Steklov egenvality 的知识中估算/ 恢复反光谱指数的反光谱问题。 由于egenvaly 值与实际估值的再折变指数是单数的, 意味着它们可以用于非破坏性测试。 一些数字例子用于反光谱问题 。