Guaranteed and robust a posteriori error estimation based on flux reconstruction for discontinuous Galerkin methods
Résumé
We propose and study a posteriori error estimates for convection--diffusion--reaction problems approximated by discontinuous Galerkin methods. Firstly, we show that accurate H(div)-conforming diffusive and convective fluxes can be reconstructed for such methods, even on meshes with hanging nodes. We next present abstract a posteriori error estimates for potential- and flux-nonconforming approximations, which are sharp since they hold for arbitrary conforming reconstructions of the discrete solution and its fluxes. In particular the classical elementwise residual can be transformed in most cases into a higher-order ``data oscillation'' term. We then derive fully computable a posteriori error estimates. In the pure diffusion case, certain parts of the estimates are robust with respect to diffusion inhomogeneities whenever diffusion-dependent weighted averages are used in the discontinuous Galerkin method. The other parts are so under some additional assumption. In the general case including singularly perturbed regimes with dominant convection or reaction, the local efficiency of the estimates in the energy norm depends on cutoff functions of the local Péclet and Damköhler numbers. The estimates become fully robust if the energy norm is augmented by a dual seminorm of the skew-symmetric part of the differential operator and a certain mesh-dependent seminorm of the jumps. Finally, numerical experiments are presented to illustrate the theoretical results.
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