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The classical Biot's theory provides the foundation of a fully dynamicporoelasticity model describing the propagation of elastic waves influid-saturated media. Multiple network poroelastic theory (MPET) takes intoaccount that the elastic matrix (solid) can be permeated by one or several($n\ge1$) superimposed interacting single fluid networks of possibly differentcharacteristics; hence the single network (classical Biot) model can beconsidered as a special case of the MPET model. We analyze the stabilityproperties of the time-discrete systems arising from second-order implicit timestepping schemes applied to the variational formulation of the MPET model andprove an inf-sup condition with a constant that is independent of all modelparameters. Moreover, we show that the fully discrete models obtained for afamily of strongly conservative space discretizations are also uniformly stablewith respect to the spatial discretization parameter. The norms in which theseresults hold are the basis for parameter-robust preconditioners.

On the stability of time-discrete dynamic multiple network poroelasticity systems arising from second-order implicit time-stepping schemes