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 implying that they can be used for nondestructive testing. Some numerical examples are provided for the inverse spectral problem.

在本文中，我们考虑在逆声散射中产生的Steklov特征值问题的数值近似。潜在的散射问题是非均质各向同性介质的问题。这些特征值已被提议用作目标特征，因为它们可以从散射数据中恢复。研究了一种Galerkin方法，其中基函数是拉普拉斯算子的Neumann本征函数。本征值和本征函数的误差估计是通过诉诸于韦尔定律证明的。我们将针对变量分离测试此方法，以验证理论收敛性。我们还考虑了根据Steklov特征值的知识估算/恢复折射率的逆光谱问题，因为特征值相对于实值折射率是单调的，这意味着它们可以用于无损检测。提供了一些反频谱问题的数值示例。