Thinh T. Kieu

Thinh T. Kieu

The nonlinear Forchheimer equations are used to describe the dynamics of fluid flows in porous media when Darcy's law is not applicable. In this article, we consider the generalized Forchheimer flows for slightly compressible fluids, and then study the expanded mixed finite element method applied to the initial boundary value problem for the resulting degenerate parabolic equation for pressure. The bounds for the solutions, time derivative and gradient of solutions are established. Utilizing the monotonicity properties of Forchheimer equation and boundedness of solutions, a {\it priori } error estimates for solution are obtained in $L^2$-norm, $L^\infty$-norm as well as for its gradient in $L^{2-a}$-norm for all $a\in (0,1)$. Optimal $L^2$-error estimates are shown for solutions under some additional regularity assumptions. Numerical results using the lowest order Raviart-Thomas mixed element confirm the theoretical analysis regarding convergence rates.

Akif Ibragimov, Thinh T. Kieu

We study the expanded mixed finite element method applied to degenerate parabolic equations with the Dirichlet boundary condition. The equation is considered a prototype of the nonlinear Forchheimer equation, a inverted to the nonlinear Darcy equation with permeability coefficient depending on pressure gradient, for slightly compressible fluid flow in porous media. The bounds for the solutions are established. In both continuous and discrete time procedures, utilizing the monotonicity properties of Forchheimer equation and boundedness of solutions we prove the optimal error estimates in $L^2$-norm for solution. The error bounds are established for the solution and divergence of the vector variable in Lebesgue norms and Sobolev norms under some additional regularity assumptions. A numerical example using the lowest order Raviart-Thomas ($RT_0$) mixed element are provided agreement with our theoretical analysis.