On 2D elliptic discontinuous Galerkin methods

We discuss the discretization using discontinuous Galerkin (DG) formulation of an elliptic Poisson problem. Two commonly used DG schemes are investigated: the original average flux proposed by Bassi and Rebay (J. Comput. Phys. 1997; 131:267) and the local discontinuous Galerkin (LDG) (SIAM J. Numer. Anal. 1998; 35:2440–2463) scheme. In this paper we expand on previous expositions (Discontinuous Galerkin Methods: Theory, Computation and Applications. Springer: Berlin, 2000; 135–146; SIAM J. Sci. Comput. 2002; 24(2):524–547; Int. J. Numer. Meth. Engng. 2003; 58(2): 1119–1148) by adopting a matrix based notation with a view to highlighting the steps required in a numerical implementation of the DG method. Through consideration of standard C0-type expansion bases, as opposed to elementally orthogonal expansions, with the matrix formulation we are able to apply static condensation techniques to improve efficiency of the direct solver when high order expansions are adopted. The use of C0-type expansions also permits the direct enforcement of Dirichlet boundary conditions through a ‘lifting’ approach where the LDG flux does not require further stabilization. In our construction we also adopt a formulation of the continuous DG fluxes that permits a more general interpretation of their numerical implementation. In particular it allows us to determine the conditions under which the LDG method provides a near local stencil. Finally a study of the conditioning and the size of the null space of the matrix systems resulting from the DG discretization of the elliptic problem is undertaken