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 stable linear system offered by wavelets are useful in dealing with the pollution effect. By using the direct approach presented in our past work [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 a standard Galerkin method, 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. In contrast, when an iterative scheme is applied to the coefficient matrix of a standard Galerkin method, 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：亥姆霍兹方程因污染效应而难以数值求解，常导致大规模病态线性系统。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空腔电磁散射问题。小波提供的高逼近阶与稀疏稳定线性系统有助于处理污染效应。通过采用我们先前工作[B. Han and M. Michelle, Appl. Comp. Harmon. Anal., 53 (2021), 270-331]中提出的直接方法，本文提出了多种在有限区间上优化的样条双正交小波。我们给出了自包含的证明，表明此类小波的张量积在适当的Sobolev空间中构成二维Riesz小波。与标准伽辽金方法的系数矩阵相比，当迭代方案应用于我们小波伽辽金方法的系数矩阵时，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，所需迭代次数实际上与小波系数矩阵的规模无关。相比之下，当迭代方案应用于标准伽辽金方法的系数矩阵时，每将每个轴的网格尺寸减半，所需迭代次数就会翻倍。得益于小波基的简单结构、可细化性质及解析表达式，算法实现也十分便捷。