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Title: A Stabilized Discretization and Robust Solvers for Poroelasticity

Abstract:
The interaction between the deformation and fluid flow in a fluid-saturated porous medium is the object of study in poroelasticity theory.  In this work, we consider a popular mixed finite-element (P1-RT0-P0) discretization of the three-field formulation of Biot’s consolidation problem, which describes linear elastic, homogeneous porous media that is saturated by an incompressible Newtonian fluid. Since this finite-element formulation does not satisfy an inf-sup condition uniformly 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. Thus, we proposed a stabilization technique that enriches the piecewise linear finite-element space of the displacement with the span of edge/face bubble functions. We have shown that for Biot’s model this does give rise to discretizations that are uniformly stable with respect to the physical parameters. We also propose a perturbation of the bilinear form, which allows for local elimination of the bubble functions and provides a uniformly stable scheme with the same number of degrees of freedom as the classical P1-RT0-P0 approach.  Additionally, we discuss robust linear solvers for this stabilized discretization of the poroelastic equations. Since the discretization is well-posed with respect to the physical and discretization parameters, it provides a framework to develop block preconditioners that are robust with respect to such parameters as well. We construct these preconditioners for the stabilized discretization and the perturbation of the stabilized discretization that leads to a smaller overall problem. Numerical results confirm the robustness of the block preconditioners. Time permitting, we also discuss a monolithic geometric multigrid method for solving the stablized discretization and compare its performance with the block preconditioners.  This is joint work with Francisco Gaspar, Xiaozhe Hu, Peter Ohm, Carmen Rodrigo, and Ludmil Zikatanov.