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Analysis of expanded mixed finite element methods for the generalized forchheimer flows of slightly compressible fluids
Author(s) -
Kieu Thinh T.
Publication year - 2016
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.21984
Subject(s) - mathematics , finite element method , norm (philosophy) , monotonic function , boundary value problem , compressibility , mathematical analysis , partial differential equation , degenerate energy levels , mixed finite element method , darcy–weisbach equation , a priori and a posteriori , porous medium , porosity , mechanics , physics , quantum mechanics , political science , philosophy , geotechnical engineering , epistemology , engineering , law , thermodynamics
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 priori error estimates for solution are obtained inL 2 ‐norm,L ∞ ‐norm as well as for its gradient inL 2 − a‐norm for all a ∈ ( 0 , 1 ) . OptimalL 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. Published 2015. Numer Methods Partial Differential Eq 32: 60–85, 2016