Discontinuous Galerkin methods for reactive transport in porous media
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Four discontinuous Galerkin (DG) methods, Oden-Baumann-Babuˇska DG (OBB-DG), Nonsymmetric Interior Penalty Galerkin (NIPG), Symmetric Interior Penalty Galerkin (SIPG) and Incomplete Interior Penalty Galerkin (IIPG), are proposed to solve the nonlinear parabolic equations arising from reactive transport and are investigated theoretically and computationally, be cause their special convergence, local conservation and approximation proper ties are attractive for parallel adaptive computation. Optimal error estimates of concentration in energy norm for the four primal DG schemes are derived. Using a parabolic lift theorem proved in this dissertation, optimal error estimates in L2(L2) and negative norms are obtained for SIPG. Convergence behaviors are conﬁrmed and further inves tigated by numerical experiments. Implementation issues such as choice of penalty parameters and physical versus reference polynomial space are ex plored. Explicit a posteriori error estimators in L2(L2) norm are established for the SIPG applied to transport with general kinetic reaction. Explicit a poste riori error estimators in linear functional and in negative norms are obtained for SIPG. Explicit a posteriori error estimators in energy norm are derived for the four DG methods. Numerical examples demonstrate that the explicit a posteriori error estimators can be computed eﬃciently, and are eﬀective in capturing local phenomena in reactive transport phenomena. Numerical re sults also conﬁrm the signiﬁcance and the necessity of adaptivity for reactive transport problems. DG methods are applied to a few benchmark cases. Test using a maze problem conﬁrms that DG handles highly varied coeﬃcients problem very well and has less numerical diﬀusion. Computation of a locally reﬁned mesh ex ample shows that DG can eﬀectively be applied to heterogeneous transport problem. Simulation of a benchmark case, the ANDRA-Couplex1 case, indi cates that the local conservation property of DG helps to reduce the accumu lated errors, and that DG is eﬀective in both the diﬀusion dominated and the advection dominated problems. Adaptive solution of a transport case with heterogeneous adsorption demonstrates the simplicity of the error indicators and the eﬀectiveness of the adaptivity based on the indicators.