A Compact Discontinuous Galerkin (CDG) Method for Elliptic Problems

We present a compact discontinuous Galerkin (CDG) method for an elliptic model problem. The problem is first cast as a system of first order equations by introducing the gradient of the primal unknown, or flux, as an additional variable. A standard discontinuous Galerkin (DG) method is then applied to the resulting system of equations. The numerical interelement fluxes are such that the equations for the additional variable can be eliminated at the element level, thus resulting in a global system that only involves the original unknown variable. The proposed method is closely related to the Local Discontinuous Galerkin (LDG) method [8], but, unlike the LDG method, the sparsity pattern of the CDG method only involves nearest neighbors. Also, unlike the LDG method, the CDG method works without stabilization for an arbitrary orientation of the element interfaces. The computation of the numerical interface fluxes for the CDG method is slightly more involved than for the LDG method, but this additional complication is clearly off-set by the increased compactness and flexibility. Compared to the BR2 method proposed in [2], which is known to be compact, the present method produces fewer non-zero elements in the matrix, is computationally more efficient, and is found to produce slightly more accurate results in the tests considered.