The Error Analysis of the Weak Galerkin Finite Element Method for Two-Dimensional Incompressible Immiscible Displacement in Porous Media

Abstract

A nonlinear system of two couple partial differential equations models for two dimensional incompressible immiscible displacement fluid in a porous medium is discussed. A sequential implicit time-stepping procedure is defined, in which the pressure and Darcy velocity of the mixture are approximated by a weak Galerkin mixed finite element method and the saturation is approximated by a weak Galerkin finite element method. Error estimate in ( and ) - norms, the stability and energy conservation are proved. Finally, a numerical experiment are presented to demonstrate the theoretical results.