# Staggered discontinuous Galerkin methods for the Helmholtz equations with large wave number

## TL;DR

This paper introduces a flexible staggered discontinuous Galerkin method for solving Helmholtz equations with large wave numbers on complex meshes, demonstrating stability, convergence, and effective handling of singular solutions.

## Contribution

It presents a novel, flux-free, and mesh-robust staggered discontinuous Galerkin approach for Helmholtz equations with large wave numbers, accommodating rough and distorted grids.

## Key findings

- Method is stable when κh is small.
- Error estimates are established in L2 norm.
- Numerical experiments confirm theoretical results.

## Abstract

In this paper we investigate staggered discontinuous Galerkin method for the Helmholtz equation with large wave number on general quadrilateral and 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 $\kappa h$ is sufficiently small, where $\kappa$ is the wave number and $h$ is the mesh size. 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.