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New stabilized discretizations for poroelasticity and the Stokes’ equations. (English) Zbl 1440.76027

Summary: In this work, we consider the popular P1-RT0-P0 discretization of the three-field formulation of Biot’s consolidation problem. Since this finite-element formulation is not uniformly stable with respect to the physical parameters, several issues arise in numerical simulations. For example, when the permeability is small with respect to the mesh size, volumetric locking may occur. To alleviate such problems, we consider a well-known stabilization technique with face bubble functions. We then design a perturbation of the bilinear form, which allows for local elimination of the bubble functions. We further prove that such perturbation is consistent and the resulting scheme has optimal approximation properties for both Biot’s model as well as the Stokes’ equations. For the former, the number of degrees of freedom is the same as for the classical P1-RT0-P0 discretization and for the latter (Stokes’ equations) the number of degrees of freedom is the same as for a P1-P0 discretization. We present numerical tests confirming the theoretical results for the poroelastic and the Stokes’ test problems.

MSC:

76D07 Stokes and related (Oseen, etc.) flows
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
76M10 Finite element methods applied to problems in fluid mechanics
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
76S05 Flows in porous media; filtration; seepage
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