Well posedness of fully coupled fracture/bulk darcy flow with XFEM

In this work we consider the coupled problem of Darcy flow in a fracture and the surrounding porous medium. The fracture is represented as a (d - 1)-dimensional interface, and it is nonmatching with the computational grid thanks to a suitable extended finite element method (XFEM) enrichment of the mixed finite element spaces. In the existing literature well posedness has been proven for the discrete problem in the hypothesis of a given solution in the fracture. This work provides theoretical results on the stability and convergence of the discrete, fully coupled problem, yielding sharp conditions on the fracture geometry and on the computational grid to ensure that the inf-sup condition is satisfied by the enriched spaces, as confirmed by numerical experiments.