Symmetric and nonsymmetric discontinuous Galerkin methods for reactive transport in porous media. (English) Zbl 1086.76043
Summary: For solving reactive transport problems in porous media, we analyze three primal discontinuous Galerkin (DG) methods with penalty, namely, symmetric interior penalty Galerkin (SIPG), nonsymmetric interior penalty Galerkin, and incomplete interior penalty Galerkin method. A cut-off operator is introduced in DG to treat general kinetic chemistry. Error estimates in \(L^{2}(H^{1})\) are established, which are optimal in \(h\) and nearly optimal in \(p\). We develop a parabolic lift technique for SIPG, which leads to \(h\)-optimal and nearly \(p\)-optimal error estimates in the \(L^{2}(L^{2})\) and negative norms. Numerical results validate these estimates. We also discuss implementation issues including penalty parameters and the choice of physical versus reference polynomial spaces.
MSC:
76M10 | Finite element methods applied to problems in fluid mechanics |
76V05 | Reaction effects in flows |
76S05 | Flows in porous media; filtration; seepage |
65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |