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IMA Journal of Numerical Analysis Advance Access published online on March 30, 2009
IMA Journal of Numerical Analysis, doi:10.1093/imanum/drn080
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© The author 2009. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Robust estimates for the approximation of the dynamic consolidation problem

Martin Sauter{dagger} and Christian Wieners{ddagger}

Fakultät für Mathematik, Universität Karlsruhe, Kaiserstrasse 12, 76128 Karlsruhe, Germany

{dagger} Email: sauter{at}math.uni-karlsruhe.de

{ddagger} Corresponding author. Email: wieners{at}math.uni-karlsruhe.de

Received on 3 August 2007. Revised on 18 November 2008.


  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.

Key Words: poro-elasticity; Biot consolidation problem; porous media; Newmark scheme; inf–sup stability; finite elements; robust estimates


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