论文标题
Helmholtz传播特征值问题的混合元素方案
A mixed element scheme of Helmholtz transmission eigenvalue problem for anisotropic media
论文作者
论文摘要
在本文中，我们研究了两个维度和三个维度的折射索引$ n（x）\ equiv 1 $的helmholtz传播特征值问题。从Cakoni，Colton和Haddar [2009]建立的非线性第四阶配方开始，通过引入一些辅助变量，我们为此问题提供了同等的混合配方，然后由有限的元素离散化。使用所提出的方案，我们严格地表明，可以预期在凸和非convex域上传输特征值的最佳收敛速率。此外，通过这种方案，我们将获得一个稀疏的广义特征值问题，即使用粗网格进行粗糙，它的大小也是如此，以至于其最小的少数实际特征值无法通过转移和反转方法来解决。我们通过使几乎所有巨大多重性的$ \ infty $特征值对矩阵大小的大幅度降低而不会降低稀疏性，从而部分克服了这一关键问题。据报道，广泛的数值示例证明了拟议方案的有效性和效率。
In this paper, we study the Helmholtz transmission eigenvalue problem for inhomogeneous anisotropic media with the index of refraction $n(x)\equiv 1$ in two and three dimension. Starting with a nonlinear fourth order formulation established by Cakoni, Colton and Haddar [2009], by introducing some auxiliary variables, we present an equivalent mixed formulation for this problem, followed up with the finite element discretization. Using the proposed scheme, we rigorously show that the optimal convergence rate for the transmission eigenvalues both on convex and nonconvex domains can be expected. Moreover, by this scheme, we will obtain a sparse generalized eigenvalue problem whose size is so demanding even with a coarse mesh that its smallest few real eigenvalues fail to be solved by the shift and invert method. We partially overcome this critical issue by deflating the almost all of the $\infty$ eigenvalue of huge multiplicity, resulting in a drastic reduction of the matrix size without deteriorating the sparsity. Extensive numerical examples are reported to demonstrate the effectiveness and efficiency of the proposed scheme.
