Staggered discontinuous Galerkin methods for the Helmholtz equations with large wave number
For computational scientists solving Helmholtz problems with large wave numbers, this method offers a flexible, flux-free approach on rough meshes, but the condition κh small limits its practical applicability for high-frequency problems.
This paper develops a staggered discontinuous Galerkin method for the Helmholtz equation with large wave numbers on general quadrilateral and polygonal meshes, proving stability and convergence under the condition that κh is sufficiently small. Numerical experiments verify the theoretical results and demonstrate the method's capability for capturing singular solutions.
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 $κh$ is sufficiently small, where $κ$ 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.