Local Discontinuous Galerkin Method for Incompressible Miscible Displacement Problem in Porous Media

In this paper, we develop local discontinuous Galerkin method for the two-dimensional coupled system of incompressible miscible displacement problem. Optimal error estimates in $$L^{\infty }(0, T; L^{2})$$L∞(0,T;L2) for concentration c, $$L^{2}(0, T; L^{2})$$L2(0,T;L2) for $$\nabla c$$∇c and $$L^{\infty }(0, T; L^{2})$$L∞(0,T;L2) for velocity $$\mathbf{u}$$u are derived. The main techniques in the analysis include the treatment of the inter-element jump terms which arise from the discontinuous nature of the numerical method, the nonlinearity, and the coupling of the models. The main difficulty is how to treat the inter-element discontinuities of two independent solution variables (one from the flow equation and the other from the transport equation) at cell interfaces. Numerical experiments are shown to demonstrate the theoretical results.