New Analysis of Galerkin FEMs for Miscible Displacement in Porous Media

The paper is concerned with optimal error estimates of classical Galerkin FEMs for the equations of incompressible miscible flows in porous media. The analysis done in the last several decades shows that classical Galerkin FEMs provide the numerical concentration of the accuracy O (τ^k + h^r+1 + h^s) in L²-norm. This analysis leads to the use of higher order finite element approximation to the pressure than that to the concentration in various numerical simulations to achieve the best rate of convergence. However, this error estimate is not optimal. The purpose of this paper is to establish the optimal L² error estimate O (τ^k + h^r+1 + h^s+1), from which one can see that the best convergence rate can be achieved by taking the same order (r = s) approximation to the concentration and pressure. Clearly Galerkin FEMs with r = s are less expensive in computation and easier for implementation. Numerical results for both two and three-dimensional models are presented to confirm our theoretical analysis.