An Analysis of the Embedded Discontinuous Galerkin Method for Second-Order Elliptic Problems

Abstract

The embedded discontinuous Galerkin methods are obtained from hybridizable discontinuous Galerkin methods by a simple change of the space of the hybrid unknown. In this paper, we consider embedded methods for second-order elliptic problems obtained from hybridizable discontinuous methods by changing the space of the hybrid unknown from discontinuous to continuous functions. This change results in a significantly smaller stiffness matrix whose size and sparsity structure coincides with those of the stiffness matrix of the statically condensed continuous Galerkin method. It is shown that this computational advantage has to be balanced against the fact that the approximate solutions for the scalar variable and its flux lose each a full order of convergence. Indeed, we prove that if polynomials of degree $k\ge1$ are used for the original hybridizable discontinuous Galerkin method, its approximations to the scalar variable and its flux converge with order $k+2$ and $k+1$, respectively, whereas those of the corresponding embedded discontinuous Galerkin method converge with orders $k+1$ and k, respectively, only. We also provide numerical results comparing the relative efficiency of the methods.