Galerkin finite element method for generalized Forchheimer equation of slightly compressible fluid in porous media

We consider the generalized Forchheimer flows for slightly compressible fluids. Using Muskat's and Ward's general form of Forchheimer equations, we describe the fluid dynamics by a nonlinear degenerate parabolic equation for the density. We study Galerkin finite elements method for the initial boundary value problem. The existence and uniqueness of the approximation are proved. The prior estimates for the solutions in $L^\infty(0,T;L^q(\Omega)), q\ge 2$, time derivative in $L^\infty(0,T;L^2(\Omega))$ and gradient in $L^\infty(0,T;W^{1,2-a}(\Omega)),$ with $a\in (0,1)$ are established. Error estimates for the density variable are derived in several norms for both continuous and discrete time procedures. Numerical experiments using backward Euler scheme confirm the theoretical analysis regarding convergence rates.