Stress and flux reconstruction in Biot's poro-elasticity problem with application to a posteriori error analysis

We derive equilibrated reconstructions of the Darcy velocity and of the total stress tensor for Biot's poro-elasticity problem. Both reconstructions are obtained from mixed finite element solutions of local Neumann problems posed over patches of elements around mesh vertices. The Darcy velocity is reconstructed using Raviart-Thomas finite elements and the stress tensor using Arnold-Winther finite elements so that the reconstructed stress tensor is symmetric. Both reconstructions have continuous normal component across mesh interfaces. Using these reconstructions, we derive a posteriori error estimators for Biot's poro-elasticity problem, and we devise an adaptive space-time algorithm driven by these estimators. The algorithm is illustrated on test cases with analytical solution, on the quarter five-spot problem, and on an industrial test case simulating the excavation of two galleries.