An efficient spectral-Galerkin approximation based on dimension reduction scheme for transmission eigenvalues in polar geometries

Abstract

In this paper, we put forward an efficient spectral-Galerkin approximation in view of dimension reduction scheme for transmission eigenvalue problem in polar geometries. Firstly, we turn the original problem into an equivalent fourth order nonlinear eigenvalue problem. Then the fourth order nonlinear eigenvalue problem is transformed into a coupled fourth order linear eigenvalue system by introducing an auxiliary Poisson equation. Secondly, based on polar coordinate transformation, we further reduce the coupled fourth order linear eigenvalue system to a series of equivalent one-dimensional eigenvalue systems. Thirdly, we derive the essential polar condition and introduce the appropriate weighted Sobolev space according to the polar condition, and establish the weak form and the corresponding discrete form. In addition, by utilizing spectral theory of compact operators, we prove the error estimates of approximation eigenvalues and eigenvectors for each one-dimensional eigenvalue system. Finally, we provide ample numerical experiments, and the numerical results show the effectiveness of the algorithm and the correctness of the theoretical results.

Introduction

The transmission eigenvalue problem, which is a boundary value problem in the bounded domain, plays an important role in the inverse scattering theory. [1], [2], [3], [4], [5], [6], [7], [8]. Since the transmission eigenvalues can be acquired from the far-field data of the scattered wave, we can estimate the material properties of the scattering object by using the transmission eigenvalues [9], [10], [11], [12], [13]. The interior transmission eigenvalue problem for the scattering of acoustic waves is: Find $k\in ℂ,\psi ,\varphi \in L^2\left(\Omega \right),\psi -\varphi \in H_0^2\left(\Omega \right)$ such that

$$\Delta \psi +k^{2}n\psi =0,\text{ in }\Omega ,$$$$\Delta \varphi +k^2\varphi =0,\text{ in }\Omega ,$$$$\psi -\varphi =0,\text{ on }\partial \Omega ,$$$$\frac{\partial \psi }{\partial \nu }-\frac{\partial \varphi }{\partial \nu }=0,\text{ on }\partial \Omega ,$$

where $\nu$ is the unit outward normal to the boundary $\partial \Omega$, $n$ is the index of refraction and $k$ is the transmission eigenvalue.

In recent years, more and more researchers begin to pay attention to the numerical calculation of transmission eigenvalues. The first numerical method was proposed in [14], which include the Argyris element, continuous and mixed finite element. Then, various finite element methods are proposed to solve the transmission eigenvalue problem [15], [16], [17], [18], [19], [20], [21]. But they are all based on the low-order finite element method. It is difficult and expensive to develop high-precision numerical solutions, especially for some high-dimensional transmission eigenvalue problems. As is known to us, the spectral method is an important numerical method for solving differential equations due to its high-order accuracy [22], [23], [24], [25], [26]. Therefore, some spectral methods are also developed for solving the transmission eigenvalues [27], [28], [29]. However, these spectral methods are mainly based on generalized eigenvalue problem related to the transmission eigenvalue problem and need to repeatedly solve a fourth order eigenvalue problem in the process of iteration, which will spend a large amount of computing time and can only calculate real eigenvalues.

Thus, we shall put forward an efficient spectral-Galerkin approximation in view of dimension reduction scheme for transmission eigenvalue problem in polar geometries. Firstly, we turn the original problem into an equivalent fourth order nonlinear eigenvalue problem. Then the fourth order nonlinear eigenvalue problem is transformed into a coupled fourth order linear eigenvalue system by introducing an auxiliary Poisson equation. Secondly, based on polar coordinate transformation, we further reduce the coupled fourth order linear eigenvalue system to a series of equivalent one-dimensional eigenvalue systems. Thirdly, we derive the essential polar condition and introduce the appropriate weighted Sobolev space according to the polar condition, and establish the weak form and the corresponding discrete form. In addition, by utilizing spectral theory of compact operators, we prove the error estimates of approximation eigenvalues and eigenvectors for each one-dimensional eigenvalue system. Finally, we provide ample numerical experiments, and the numerical results show the effectiveness of the algorithm and the correctness of the theoretical results.

We briefly introduce the content in the remainder of this paper. In Section 2, we derive the dimension reduction scheme and its Galerkin approximation. In Section 3, we prove error estimations of approximate eigenvalues and eigenfunctions. In Section 4, we represent some details for an effective implementation of the algorithm. In Section 5, we provide several numerical examples to illustrate the accuracy and efficiency of our algorithm. Finally, in Section 6, we make a few concluding remarks.