Ill-conditioning of ®nite element poroelasticity equations Massimiliano Ferronato * , Giuseppe Gambolati, Pietro Teatini DMMMSA ± Department of Mathematical Methods and Models for Scienti®c Applications, University of Padova, Via Belzoni 7, 35131 Padova, Italy Received 15 March 2000 Abstract The solution to Biot's coupled consolidation theory is usually addressed by the ®nite element FE) method thus obtaining a system of ®rst-order dierential equations which is integrated by the use of an appropriate time marching scheme. For small values of the time step the resulting linear system may be severely ill-conditioned and hence the solution can prove quite dicult to achieve. Under such conditions ecient and robust projection solvers based on Krylov's subspaces which are usually recommended for non-symmetric large size problems can exhibit a very slow convergence rate or even fail. The present paper investigates the correlation between the ill-conditioning of FE poroelasticity equations and the time integration step Dt. An empirical relation is provided for a lower bound Dt crit of Dt below which ill-conditioning may suddenly occur. The critical time step is larger for soft and low permeable porous media discretized on coarser grids. A limiting value for the rock stiness is found such that for stier systems there is no ill-conditioning irrespective of Dt however small, as is also shown by several numerical examples. Finally, the de®nition of a dierent Dt crit as suggested by other authors is reviewed and discussed. Ó 2001 Elsevier Science Ltd. All rights reserved. Keywords: Finite elements; Coupled poroelasticity; Ill-conditioning; Critical time step; Convergence rate; Projective solvers 1. Introduction The time-dependent distribution of displacement and ¯uid pore pressure in porous media was ®rst mathematically described by Biot 1941). Biot's consolidation theory couples the elastic equilibrium equations with a continuity or mass balance equation which may be solved under appropriate boundary and initial ¯ow and loading conditions. The consolidation problem is usually solved in space by a ®nite element FE) technique giving rise to a system of ®rst-order dierential equations. The solution to these equations is typically addressed by an appropriate time marching scheme. The discretization in the time domain may require variable time steps which may change by several orders of magnitude during the analysis. As a matter of fact, in the early phase International Journal of Solids and Structures 38 2001) 5995±6014 www.elsevier.com/locate/ijsolstr * Corresponding author. Tel.: +39-049-8275929; fax: +39-049-8725995. E-mail address: ferronat@dmsa.unipd.it M. Ferronato). 0020-7683/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII:S0020-768300)00352-8