Adaptive poromechanics computations based on a posteriori error estimates for fully mixed formulations of Biot's consolidation model

Abstract : This paper is concerned with the analysis of coupled mixed finite element methods applied to the Biot's consolidation model. We consider two mixed formulations that use the stress tensor and Darcy velocity as primary variables as well as the displacement and pressure. The first formulation is with a symmetric stress tensor while the other enforces the symmetry of the stress weakly through the introduction of a Lagrange multiplier. The well-posedness of the two formulations is shown through Galerkin's method and suitable a priori estimates. The two formulations are then discretized with the backward Euler scheme in time and with two mixed finite elements in space. We present next a general and unified a posteriori error analysis which is applicable for any flux- and stress-conforming discretization. Our estimates are based on $H^{1}(\Omega)$-conforming reconstruction of the pressure and a suitable $\left[H^{1}(\Omega)\right]^{d}$-conforming reconstruction of the displacement; both are continuous and piecewise affine in time. These reconstructions are used to infer a guaranteed and fully computable upper bound on the energy-type error measuring the differences between the exact and the approximate pressure and displacement. The error components resulting from the spatial and the temporal discretization are distinguished. They are then used to design an adaptive space--time algorithm. Numerical experiments illustrate the efficiency of our estimates and the performance of the adaptive algorithm.