An optimal‐order error estimate on an H¹‐Galerkin mixed method for a nonlinear parabolic equation in porous medium flow

Abstract

We present an H 1 -Galerkin mixed finite element method for a nonlinear parabolic equation, which models a compressible fluid flow process in subsurface porous media. The method possesses the advantages of mixed finite element methods while avoiding directly inverting the permeability tensor, which is important especially in a low permeability zone. We conducted theoretical analysis to study the existence and uniqueness of the numerical solutions of the scheme and prove an optimal-order error estimate for the method. I. A MATHEMATICAL MODEL FOR POROUS MEDIUM FLOWThe objective of this article is to present and analyze an H 1 -Galerkin mixed finite element method for a nonlinear parabolic equation in subsurface porous medium flow. Mathematical models used to describe porous medium flow processes in petroleum reservoir simulation, groundwater contaminant transport, and other applications lead to a coupled system of time-dependent nonlinear partial differential equations.Let c(x, t) be the concentration (or mass fraction) of an invading fluid or a concerned solute or solvent, and let p(x, t) and u(x, t) be the pressure and Darcy velocity of the fluid mixture. The

Unconditional Convergence and Optimal Error Estimates of a Galerkin-Mixed FEM for Incompressible Miscible Flow in Porous Media

“…The above system was investigated extensively in the last several decades [3,8,17,18,20] due to its wide applications in various engineering areas, such as reservoir simulations and exploration of underground water, oil and gas. Existence of weak solutions of the system was obtained by Feng [20] for the 2D model and by Chen and Ewing [6] for the 3D problem.…”

Error Estimates on Hybridizable Discontinuous Galerkin Methods for Parabolic Equations with Nonlinear Coefficients

“…The existence and the uniqueness (which depend on the Lipschitz condition that will be presented in the next section and the choices of the approximation spaces and the stabilization parameter ) on the solution for the system of equations (7a) to (7f) can be found in [28,29].…”

A fourth‐order H1‐Galerkin mixed finite element method for Kuramoto–Sivashinsky equation

“…[40] and to a nonlinear parabolic equation in porous medium flow in Ref. [41]. For more applications of this mixed formulation for pseudo-hyperbolic and for a class of heat transport equations, see [42,43].…”