An H(div)-Conforming Finite Element Method for the Biot Consolidation Model

Abstract

An $H$(div)-conforming finite element method for the Biot's consolidation model is developed, with displacements and fluid velocity approximated by elements from BDM_$k$ space. The use of $H$(div)-conforming elements for flow variables ensures the local mass conservation. In the $H$(div)-conforming approximation of displacement, the tangential components are discretised in the interior penalty discontinuous Galerkin framework, and the normal components across the element interfaces are continuous. Having introduced a spatial discretisation, we develop a semi-discrete scheme and a fully discrete scheme, prove their unique solvability and establish optimal error estimates for each variable.