Robust estimates for the approximation of the dynamic consolidation problem

Abstract

We consider stable discretizations in time and space for the linear dynamic consolidation problem describing wave propagation in a porous solid skeleton that is fully saturated with an incompressible fluid. Introducing the hydrostatic pressure, the flow problem is described by Darcy's law. In particular, we discuss the case of nearly-impermeable solids, which requires inf-sup stable discretizations in space for the limiting saddle point problem. Together with an (implicit) Newmark discretization in time, we derive convergence estimates for the fully-discrete scheme that are robust with respect to the coupling parameter of fluid and solid.