A Posteriori Error Estimation for Discontinuous Galerkin Finite Element Approximation

It is shown that the interelement discontinuities in a discontinuous Galerkin finite element approximation are subordinate to the error measured in the broken $H^1$-seminorm. One consequence is that the DG-norm of the error is equivalent to the broken energy seminorm. Computable a posteriori error bounds are obtained for the error measured in both the DG-norm and the broken energy seminorm and are shown to be efficient in the sense that they also provide lower bounds up to a constant and higher order data oscillation terms. The estimators are completely free of unknown constants and provide guaranteed numerical bounds for the error.