A weighted discontinuous Galerkin method for diffusion equations with non-smooth coefficients on distorted polygonal meshes

Abstract

The elliptic equations with heterogeneous and anisotropic diffusion coefficients play an important role in many physical applications such as radiation hydrodynamics and reservoir simulations. Usually the distorted meshes are considered since the 解 of the diffusion part involves the hydro part. The discontinuous Galerkin (DG) method is an important class of methods in computational mathematics. Thefeatures of DG methods can be implemented flexibly on large deformation quadrilateral meshes anddiscontinuous coefficients. In this study, we present a weighted discontinuous Galerkin (WDG) method for numerically solving the elliptic 问题 with non-smooth coefficients. The WDG method is based on the symmetric interior penalty technique. We give the proof of the coercivity and continuity of bilinear forms in the WDG scheme. The convergence analysis for the energy norm is presented based on the coercivity and continuity. Next, we prove an error estimate in the $L^2$ norm based on the duality argument. Numerical examples demonstrate the validity of the WDG method for elliptic problems with discontinuous and anisotropic diffusion coefficients. Some types of distorted meshes such as random, sinusoidal, Shestakov, and Z meshes are used.