A New Direct Discontinuous Galerkin Method with Symmetric Structure for Nonlinear Diffusion Equations

Abstract

In this paper we continue the study of discontinuous Galerkin nite element methods for nonlinear diusion equations following the direct discontinuous Galerkin (DDG) methods for diusion problems [17] and the direct discontinuous Galerkin (DDG) methods for diusion with interface corrections [18]. We introduce a numerical ux for the test function, and obtain a new direct discontinuous Galerkin method with symmetric structure. Second order derivative jump terms are included in the numerical ux formula and explicit guidelines for choosing the numerical ux are given. The constructed scheme has a symmetric property and an optimalL 2 (L 2 ) error estimate is obtained. Numerical examples are carried out to demonstrate the optimal (k + 1)th order of accuracy for the method with P k polynomial approximations for both linear and nonlinear problems, under one-dimensional and two-dimensional settings.