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 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 propose a stabilization technique that enriches the piecewise linear finite-element space of the displacement with the span of edge/face bubble functions. We show 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. We prove optimal stability and error estimates for this discretization. Finally, we show that this scheme can also be successfully applied to Stokes' equations, yielding a discrete problem with optimal approximation properties and with minimum number of degrees of freedom (equivalent to a P1-P0 discretization). Numerical tests confirm the theory for both poroelastic and Stokes' test problems.

\bfA \bfb \bfs \bft \bfr \bfa \bfc \bft. In this paper, we present block preconditioners for a stabilized discretization of the poroelastic equations developed in [C.

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In this work, we consider two discretizations of the three-field formulation of Biot's consolidation problem. They employ the lowestorder mixed finite elements for the flow (Raviart-Thomas-Nédélec elements for the Darcy velocity and piecewise constants for the pressure) and are stable with respect to the physical parameters. The difference is in the mechanics: one of the discretizations uses Crouzeix-Raviart nonconforming linear elements; the other is based on piecewise linear elements stabilized by using face bubbles, which are subsequently eliminated. The numerical solutions obtained from these discretizations satisfy mass conservation: the former directly and the latter after a simple postprocessing.

We consider monolithic algebraic multigrid (AMG) algorithms for the solution of block linear systems arising from multiphysics simulations. While the multigrid idea is applied directly to the entire linear system, AMG operators are constructed by leveraging the matrix block structure. In particular, each block corresponds to a set of physical unknowns and physical equations. Multigrid components are constructed by first applying existing AMG procedures to matrix sub-blocks. The resulting AMG sub-components are then composed together to define a monolithic AMG preconditioner. Given the problem-dependent nature of multiphysics systems, different blocking choices may work best in different situations, and so software flexibility is essential. We apply different blocking strategies to systems arising from resistive magnetohydrodynamics in order to demonstrate the associated trade-offs.