## Convergence analysis of a new mixed finite element method for Biot’s consolidation model.(English)Zbl 1350.74024

Summary: 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 in $$L^\infty(L^2)$$ when the constrained-specific storage coefficient $$c_0$$ is strictly positive and in the weaker $$L^2(L^2)$$ norm when $$c_0$$ is nonnegative. We also present some of our numerical results.

### MSC:

 74S05 Finite element methods applied to problems in solid mechanics 74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) 76S05 Flows in porous media; filtration; seepage 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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