A Staggered Discontinuous Galerkin Method of Minimal Dimension on Quadrilateral and Polygonal Meshes

Abstract

In this paper, we first propose and analyze a locally conservative, lowest order staggered discontinuous Galerkin method of minimal dimension on general quadrilateral/polygonal meshes for elliptic problems. The method can be flexibly applied to rough grids such as the highly distorted trapezoidal grid, and both $h$ perturbation and $h^2$ perturbation of the smooth grids. Optimal convergence rates for both the potential and vector variables are achieved for smooth solutions. On the other hand, the lowest order method can be particularly useful for computing rough solutions. We provide a priori error analysis for problems with low regularity. Next, adaptive mesh refinement is an attractive tool for general meshes due to their flexibility and simplicity in handling hanging nodes. Therefore, we derive a simple residual-type error estimator on the $L^2$ error in vector variable, and the reliability and efficiency of the proposed error estimator are proved. Numerical results indicate that optimal convergence can be achieved for both the potential and vector variables, and the singularity can be well-captured by the proposed error estimator.