  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.
