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

Rita Riedlbeck (1, 2) , Daniele Di Pietro (2) , Alexandre Ern (3, 4) , Sylvie Granet (1) , Kyrylo Kazymyrenko (1)

1 IMSIA - UMR 9219 - Institut des Sciences de la mécanique et Applications industrielles
2 IMAG - Institut Montpelliérain Alexander Grothendieck
3 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique
4 SERENA - Simulation for the Environment: Reliable and Efficient Numerical Algorithms

We derive equilibrated reconstructions of the Darcy velocity and of the total stress ten-sor 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.

Mots clés: Arnold-Winther finite element space; a posteriori error estimate; equilibrated stress reconstruction; Biot's poro-elasticity problem

Domaines: Analyse numérique [math.NA]

Rita Riedlbeck, Daniele Di Pietro, Alexandre Ern, Sylvie Granet, Kyrylo Kazymyrenko. Stress and flux reconstruction in Biot's poro-elasticity problem with application to a posteriori error analysis. Computers & Mathematics with Applications, 2017, 73 (7), pp.1593-1610. ⟨10.1016/j.camwa.2017.02.005⟩. ⟨hal-01366646v2⟩