Mixed finite element methods for compressible miscible displacement in porous media

Abstract

A differential system describing compressible miscible displacement in a porous medium is given. The concentration equation is treated by a Galerkin method and the pressure equation is treated by a parabolic mixed finite element method. Optimal-order estimates in L 2 {L^2} and almost optimal-order estimates in L ∞ {L^\infty } are obtained for the errors in the approximate solutions under the condition that h p 2 k + 2 ( log ⁡ h c − 1 ) 1 / 2 → 0 h_p^{2k + 2}{(\log h_c^{ - 1})^{1/2}} \to 0 . This condition is much weaker than one given earlier by Douglas and Roberts for the same model. Furthermore, we obtain the L ∞ ( L 2 ( Ω ) ) {L^\infty }({L^2}(\Omega )) -estimates for the time-derivatives of the concentration and the pressure, which were not given by the above authors. In addition, we also consider newer mixed spaces in two or three dimensions.