Adaptive strategies based on equilibration for the Biot equations

Abstract

The hypercircle theorem leads to a posteriori error estimates for the three-field variational formulation of the Biot problem involving displacements, total pressure and fluid pressure. Based thereon, adaptive strategies can be constructed. In particular, the error estimator is derived on the basis of H(div)-conforming reconstructions of stress and flux approximations while the symmetry of the reconstructed stress is allowed to be satisfied only weakly. The reconstructions can therefore be performed locally on a set of vertex patches. The local nature of the reconstruction leads to two different possible adaptive strategies. In this paper, we review these two different adaptive strategies for the Biot problem. Numerical experiments underline the differences between the two strategies.