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Convergence analysis of a new mixed finite element method for Biot's consolidation model
Author(s) -
Yi SonYoung
Publication year - 2014
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.21865
Subject(s) - mathematics , poromechanics , biot number , finite element method , mixed finite element method , consolidation (business) , mathematical analysis , cauchy stress tensor , elasticity (physics) , norm (philosophy) , mechanics , porous medium , physics , materials science , geotechnical engineering , accounting , porosity , political science , law , engineering , business , composite material , thermodynamics
In this article, we propose a mixed finite element method for the two‐dimensional Biot's consolidation model of poroelasticity. The new mixed formulation presented herein uses the total stress tensor and fluid flux as primary unknown variables as well as the displacement and pore pressure. This method is based on coupling two mixed finite element methods for each subproblem: the standard mixed finite element method for the flow subproblem and the Hellinger–Reissner formulation for the mechanical subproblem. Optimal a‐priori error estimates are proved for both semidiscrete and fully discrete problems when the Raviart–Thomas space for the flow problem and the Arnold–Winther space for the elasticity problem are used. In particular, optimality in the stress, displacement, and pressure has been proved inL ∞ ( L 2 ) when the constrained‐specific storage coefficientc 0is strictly positive and in the weakerL 2 ( L 2 ) norm whenc 0is nonnegative. We also present some of our numerical results.Copyright © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1189–1210, 2014