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：Helmholtz 等式由于污染效应而难以在数字上解决数字问题,因为污染效应往往导致极不完善的线性系统。在本文中,我们展示了一种高顺序的波子波浪加列尔金(Galerkin) 方法,以数字方式解决以 2D Helmholtz 等式为模型的大型孔洞问题。波子提供的高近似顺序和稀疏稳定的线性系统对于处理污染效应是有用的。与标准加勒金方法的系数矩阵相比,当对我们的波状方法的系数矩阵应用时,加勒金(Appl.comp. Harmon. Anal., 53 (2021), 270-331),我们展示了多种最优化的螺旋双向双向双向波波波波波波子波子阵列(Spressline striple) 方法的表达方式要更短得多。在固定的平流基底基质结构中, 需要的是固定基数的比值。</s>