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In this paper we develop adaptive iterative coupling schemes for the Biotsystem modeling coupled poromechanics problems. We particularly consider thespace-time formulation of the fixed-stress iterative scheme, in which we firstsolve the problem of flow over the whole space-time interval, then exploitingthe space-time information for solving the mechanics. Two commondiscretizations of this algorithm are then introduced based on two coupledmixed finite element methods in-space and the backward Euler scheme in-time.Therefrom, adaptive fixed-stress algorithms are build on conformingreconstructions of the pressure and displacement together with equilibratedflux and stresses reconstructions. These ingredients are used to derive aposteriori error estimates for the fixed-stress algorithms, distinguishing thedifferent error components, namely the spatial discretization, the temporaldiscretization, and the fixed-stress iteration components. Precisely, at theiteration $k\geq 1$ of the adaptive algorithm, we prove that our estimate givesa guaranteed and fully computable upper bound on the energy-type errormeasuring the difference between the exact and approximate pressure anddisplacement. These error components are efficiently used to design adaptiveasynchronous time-stepping and adaptive stopping criteria for the fixed-stressalgorithms. Numerical experiments illustrate the efficiency of our estimatesand the performance of the adaptive iterative coupling algorithms.

Adaptive asynchronous time-stepping, stopping criteria, and a posteriori  error estimates for fixed-stress iterative schemes for coupled poromechanics  problems