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We consider the generalized Forchheimer flows for slightly compressiblefluids. Using Muskat's and Ward's general form of Forchheimer equations, wedescribe the fluid dynamics by a nonlinear degenerate parabolic equation forthe density. We study Galerkin finite elements method for the initial boundaryvalue 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. Errorestimates for the density variable are derived in several norms for bothcontinuous and discrete time procedures. Numerical experiments using backwardEuler scheme confirm the theoretical analysis regarding convergence rates.

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