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

2015/08/02 by Thinh Kieu, Kieu, Thinh
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Differential Equations and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.1508.00294

openalex publication_date 2015/08/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the generalized Forchheimer flows for slightly compressible\nfluids. Using Muskat's and Ward's general form of Forchheimer equations, we\ndescribe the fluid dynamics by a nonlinear degenerate parabolic equation for\nthe density. We study Galerkin finite elements method for the initial boundary\nvalue problem. The existence and uniqueness of the approximation are proved.\nThe prior estimates for the solutions in L^\∞(0,T;L^q(\Ω)), q\≥ 2,\ntime derivative in L^\∞(0,T;L²(\Ω)) and gradient in\nL^\∞(0,T;W^1,2-a(\Ω)), with a\∈ (0,1) are established. Error\nestimates for the density variable are derived in several norms for both\ncontinuous and discrete time procedures. Numerical experiments using backward\nEuler scheme confirm the theoretical analysis regarding convergence rates.\n

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