Improved energy norm a posteriori error estimation based on flux reconstruction for discontinuous Galerkin methods
Résumé
We propose and study a new approach to residual a posteriori error estimation in the discontinuous Galerkin finite element method. The main idea, which consists of constructing an H(div)-conforming Raviart-Thomas flux on the basis of the conservative discontinuous Galerkin side fluxes, is first exposed for a pure diffusion second-order elliptic problem. In this case, the classical elementwise residual can be transformed into a higher-order term (sometimes considered separately and called "data oscillation term"), thus fully taking advantage of the spectral degrees of freedom within each element available in the discontinuous Galerkin method. Moreover, the classical estimator based on normal gradient jumps is simultaneously replaced by a comparison of the original and reconstructed diffusive fluxes. Finally, our error bound consists of one last estimator which measures the nonconformity of the actual discrete solution by comparing it to its so-called Oswald interpolate. In the second part of the paper, we extend our results to convection-diffusion-reaction problems, where we introduce an additional convective flux reconstruction. Our estimators are based on an abstract upper bound which is sharp since it is established for arbitrary conforming reconstructions of the discrete solution itself and of its diffusive and convective fluxes. They yield a guaranteed upper bound since all constants are evaluated, are locally efficient, represent local lower bounds of the classical residual estimators, and numerical examples presented at the end of the paper confirm their accuracy and robustness. Incidentally, the H(div)-conforming Raviart-Thomas diffusive and convective flux reconstructions are of independent interest.
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