A nonconforming finite element method for the Biot’s consolidation model in poroelasticity. (English) Zbl 1381.76175
Summary: A stable finite element scheme that avoids pressure oscillations for a three-field Biot’s model in poroelasticity is considered. The involved variables are the displacements, fluid flux (Darcy velocity), and the pore pressure, and they are discretized by using the lowest possible approximation order: Crouzeix-Raviart finite elements for the displacements, lowest order Raviart-Thomas-Nédélec elements for the Darcy velocity, and piecewise constant approximation for the pressure. Mass-lumping technique is introduced for the Raviart-Thomas-Nédélec elements in order to eliminate the Darcy velocity and, therefore, reduce the computational cost. We show convergence of the discrete scheme which is implicit in time and use these types of elements in space with and without mass-lumping. Finally, numerical experiments illustrate the convergence of the method and show its effectiveness to avoid spurious pressure oscillations when mass lumping for the Raviart-Thomas-Nédélec elements is used.
MSC:
| 76M10 | Finite element methods applied to problems in fluid mechanics |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
| 74F10 | Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) |
| 76S05 | Flows in porous media; filtration; seepage |
Keywords:
nonconforming finite elements; stable discretizations; monotone discretizations; poroelasticityReferences:
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