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Guaranteed and robust discontinuous Galerkin a posteriori error estimates for convection-diffusion-reaction problems. (English) Zbl 1190.65165
Authors’ abstract: We propose and study a posteriori error estimates for convection-diffusion-reaction problems with inhomogeneous and anisotropic diffusion approximated by weighted interior-penalty discontinuous Galerkin methods. Our twofold objective is to derive estimates without undetermined constants and to analyze carefully the robustness of the estimates in singularly perturbed regimes due to dominant convection or reaction.
We first derive locally computable estimates for the error measured in the energy (semi)norm. These estimates are evaluated using $$\mathbf H(\text{div},\varOmega)$$-conforming diffusive and convective flux reconstructions, thereby extending the previous work on pure diffusion problems. The resulting estimates are semi-robust in the sense that local lower error bounds can be derived using suitable cutoff functions of the local Péclet and Damköhler numbers.
Fully robust estimates are obtained for the error measured in an augmented norm consisting of the energy (semi)norm, a dual norm of the skew-symmetric part of the differential operator, and a suitable contribution of the interelement jumps of the discrete solution. Numerical experiments are presented to illustrate the theoretical results.

##### MSC:
 65N15 Error bounds for boundary value problems involving PDEs 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 76S05 Flows in porous media; filtration; seepage 35J25 Boundary value problems for second-order elliptic equations 35B25 Singular perturbations in context of PDEs
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