The Helmholtz equation is challenging to solve numerically due to the pollution effect, which often results in a huge ill-conditioned linear system. In this paper, we present a high order wavelet Galerkin method to numerically solve an electromagnetic scattering from a large cavity problem modeled by the 2D Helmholtz equation. The high approximation order and the sparse linear system with uniformly bounded condition numbers offered by wavelets are useful in reducing the pollution effect. Using the direct approach in [B. Han and M. Michelle, Appl. Comp. Harmon. Anal., 53 (2021), 270-331], we present various optimized spline biorthogonal wavelets on a bounded interval. We provide a self-contained proof to show that the tensor product of such wavelets forms a 2D Riesz wavelet in the appropriate Sobolev space. Compared to the coefficient matrix of the finite element method (FEM), when an iterative scheme is applied to the coefficient matrix of our wavelet Galerkin method, much fewer iterations are needed for the relative residuals to be within a tolerance level. Furthermore, for a fixed wavenumber, the number of required iterations is practically independent of the size of the wavelet coefficient matrix, due to the uniformly bounded small condition numbers of such wavelets. In contrast, when an iterative scheme is applied to the FEM coefficient matrix, the number of required iterations doubles as the mesh size for each axis is halved. The implementation can also be done conveniently thanks to the simple structure, the refinability property, and the analytic expression of our wavelet bases.

翻译：亥姆霍兹方程由于污染效应而难以数值求解，常导致生成大型病态线性系统。本文提出一种高阶小波伽辽金方法，用于数值求解由二维亥姆霍兹方程建模的大型空腔电磁散射问题。小波的高逼近阶、稀疏线性系统及其一致有界条件数有助于削弱污染效应。借助[B. Han and M. Michelle, Appl. Comp. Harmon. Anal., 53 (2021), 270-331]中的直接方法，我们给出了有界区间上多种优化的样条双正交小波。通过自包含的证明，论证了此类小波张量积在适当的索伯列夫空间中构成二维Riesz小波。与有限元法（FEM）系数矩阵相比，当对小波伽辽金方法系数矩阵应用迭代方案时，相对残差达到容差水平所需的迭代次数显著减少。此外，对于固定波数，由于此类小波具有一致有界的小条件数，所需迭代次数实际上与小波系数矩阵的尺寸无关。相比之下，当迭代方案应用于FEM系数矩阵时，每个轴向上网格尺寸减半，所需迭代次数则翻倍。得益于小波基的简单结构、可细化性质及解析表达式，实现过程也较为便捷。