由于在各向异性材料的非破坏性测试中的重要性，传输特征值问题已经成为逆散射理论领域里的一个新的研究课题。然而，传输特征值问题既是非自共轭的，也是非椭圆的，一些标准的偏微分方程理论并不能被直接地应用。因此，关于传输特征值问题的研究是一个有趣而且具有挑战的课题。由于物质材料的性质并不能够由散射场的远场模式唯一确定，那么怎样去估计物质材料的性质是一个很关紧的问题。最近，一种新的方法被提出，该方法是利用传输特征值去估计物质材料的性质，但是该方法需要有一个有效而且强劲的数值方法去计算传输特征值。因此，本项目拟从三个方面对传输特征值问题的高效数值计算方法作系统深入研究：（1）对各种不同的区域分别在二维和三维情况下提出一种有效的谱Galerkin方法计算传输特征值；（2）对每种数值方法给出误差估计，并从理论和数值试验两个方面证明算法的有效性；（3）将该方法用于物质材料性质的估计。

The transmission eigenvalue problem is a new research topic in the area of inverse scattering theory due to its importance in non-destructive testing of anisotropic materials. However, the transmission eigenvalue problem is non-self-adjoint and non-elliptic. The standard theory of partial differential equations can't be applied directly. Due to this fact, the study of transmission eigenvalues is an interesting, but challenging topic. Since the material properties is not uniquely determined from the far field pattern of the scattered field, then how to estimate the properties of materials is a very important problem. Resently, a new method is presented and used to estimate the material properties by using transmission eigenvalues. But the method rests on having an efficient and robust method for computing transmission eigenvalues. Thus, the project attempts to make a systematic and deep research on high effective numerical methods for transmission eigenvalue problems from three aspects: (1) To develop an efficient Spectral-Galerkin method to compute transmission eigenvalues in 2-D and 3-D cases for different domains. (2) To provide an error estimation for every numerical method and prove the efficiency of the algorithm by theory and numerical experiments. (3) To estimate material properties by using this method.