In this paper we investigate staggered discontinuous Galerkin method for the Helmholtz equation with large wave number on general polygonal meshes. The method is highly flexible by allowing rough grids such as the trapezoidal grids and highly distorted grids, and at the same time, is numerical flux free. Furthermore, it allows hanging nodes, which can be simply treated as additional vertices. By exploiting a modified duality argument, the stability and convergence can be proved under the condition that $\kappa h\le C_0{\left(\frac{1}{\kappa }\right)}^{\frac{1}{m+1}}$, where $m$ is the polynomial order, $\kappa$ is the wave number, $h$ is the mesh size and $C_0$ is a positive constant independent of $\kappa ,h$. Error estimates for both the scalar and vector variables in $L^2$ norm are established. Several numerical experiments are tested to verify our theoretical results and to present the capability of our method for capturing singular solutions.

在本文中，我们研究了一般多边形网格上具有大波数的Helmholtz方程的交错不连续Galerkin方法。该方法通过允许使用粗糙的网格（例如梯形网格和高度扭曲的网格）而具有高度的灵活性，并且同时没有数值通量。此外，它允许悬挂节点，可以简单地将其视为附加顶点。通过利用修正的对偶参数，可以在以下条件下证明稳定性和收敛性：$\kappa H\le C_0{（\frac{\mathrm{1个}}{\kappa }）}^{\frac{\mathrm{1个}}{米+\mathrm{1个}}}$，在哪里 $米$ 是多项式阶数 $\kappa$ 是波数， $H$ 是网眼尺寸和 $C_0$ 是一个独立于 $\kappa ，H$。标量和向量变量的误差估计${\mathrm{大号}}^2$建立规范。测试了几个数值实验，以验证我们的理论结果并展示我们的方法捕获奇异解的能力。