The expanded mixed finite element method for generalized Forchheimer flows in porous media
Provides rigorous error analysis for a numerical method applied to nonlinear Forchheimer flows, which is important for porous media simulations but represents an incremental theoretical extension.
The paper develops optimal error estimates for the expanded mixed finite element method applied to degenerate parabolic equations modeling Forchheimer flows in porous media, with numerical validation using RT0 elements.
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.