Analysis and preconditioning of parameter-robust finite element methods for Biot's consolidation model

Abstract

In this paper we consider a three-field formulation of the Biot model which has the displacement, the total pressure, and the pore pressure as unknowns. For parameter-robust stability analysis, we first show a priori estimates of the continuous problem with parameter-dependent norms. Then we study finite element discretizations which provide parameter-robust error estimates and preconditioners. For finite element discretizations we consider standard mixed finite element as well as stabilized methods for the Stokes equations, and the complete error analysis of semidiscrete solutions is given. Abstract forms of parameter-robust preconditioners are investigated by the operator preconditioning approach. The theoretical results are illustrated with numerical experiments.