Finite Element Center

In this paper we investigate the relationship between the continuous and the discontinuous Galerkin methods for elliptic problems. In particular, we show that the continuous Galerkin method can be interpreted as the limit of a discontinuous Galerkin method when a stabilization parameter tends to infinity. Based on this observation we derive a method for computing a conservative approximation of the flux on the the boundary of each element for the continuous Galerkin method. The conservative flux is then obtained by actually computing the limit of the natural conservative flux provided by the discontinuous Galerkin method. We prove existence, uniqueness, and optimal order error estimates. Finally, we illustrate our results by a few numerical examples