A model of fracture for elliptic problems with flux and solution jumps

A new model of fracture for elliptic problems combining flux and solution jumps as immersed boundary conditions is proposed and proved to be well-posed. An application of this model to the flow in fractured porous media is also proposed including the cases of “impermeable fracture” and “fully permeable fracture” satisfying the so-called “cubic law”, as well as intermediate cases. A finite volume scheme on general polygonal meshes is built to solve such problems. Since no unknown is required at the fracture interface, the scheme is as cheap as standard schemes for the same problems without fault. The convergence of the scheme can be proved to the weak solution of the problem. With weak regularity assumptions, we also establish for the discrete H¹₀ and L² norms some error estimates in $\text{O}\text{(h)}$, where h is the maximum diameter of the control volumes of the mesh. To cite this article: Ph. Angot, C. R. Acad. Sci. Paris, Ser. I 337 (2003).