The discontinuous Galerkin method with higher degree finite difference compatibility conditions and arbitrary local and global basis functions

Abstract

This paper focuses on the discontinuous Galerkin (DG) method in which the compatibility condition on the mesh skeleton and Dirichlet boundary condition on the outer boundary are enforced with the help of one-dimensional finite difference (FD) rules, while in the standard approach those conditions are satisfied by the penalty constraints. The FD rules can be of arbitrary degree and in this paper the rules are applied up to fourth degree. It is shown that the method presented in this paper gives better results in comparison to the standard version of the DG method. The method is based on discontinuous approximation, which means that it can be constructed using arbitrary local basis functions in each finite element. It is quite easy to incorporate some global basis functions in the approximation field and this is also shown in the paper. The paper is illustrated with a couple of two-dimensional examples.