Mathematical Problems in Mechanics/Numerical Analysis
A model of fracture for elliptic problems with flux and solution jumps
Comptes Rendus. Mathématique, Volume 337 (2003) no. 6, pp. 425-430.

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 𝒪(h), where h is the maximum diameter of the control volumes of the mesh.

Pour des problèmes elliptiques, un nouveau modèle de fracture combinant des sauts de la solution et du flux comme conditions aux limites immergées est proposé et on montre qu'il est bien posé. On propose également une application de ce modèle à l'écoulement dans des milieux poreux fissurés incluant les cas de « fracture imperméable » et de « fracture totalement perméable » satisfaisant la « loi cubique », ainsi que des cas intermédiaires. Un schéma en volumes finis est construit pour résoudre de tels problèmes sur des maillages polygonaux généraux. Comme aucune inconnue n'est nécessaire sur l'interface de fracture, ce schéma est aussi économique que des schémas standards pour résoudre les mêmes problèmes sans faille. On peut prouver la convergence de ce schéma vers la solution faible du problème. De plus, avec des hypothèses faibles de régularité, on établit pour la norme discrète H10 et pour la norme L2 des estimations d'erreur en 𝒪(h), où h désigne le pas du maillage, i.e. le diamètre maximum des volumes finis du maillage.

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DOI: 10.1016/S1631-073X(03)00300-5
Angot, Philippe 1

1 Université de la Méditerranée Aix-Marseille II, CMI–LATP, UMR CNRS 6632, 39, rue F. Joliot Curie, 13453 Marseille cedex 13, France
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Angot, Philippe. A model of fracture for elliptic problems with flux and solution jumps. Comptes Rendus. Mathématique, Volume 337 (2003) no. 6, pp. 425-430. doi : 10.1016/S1631-073X(03)00300-5. http://www.numdam.org/articles/10.1016/S1631-073X(03)00300-5/

[1] Ph. Angot, Mathematical and numerical modelling for a fictitious domain method with jump and penalized immersed boundary conditions, Preprint, Thèse HDR, Univ. Méditerranée Aix-Marseille II, sept. 1998

[2] Angot, Ph. Finite volume methods for non smooth solution of diffusion models; application to imperfect contact problems, Proc. 4th Int. Conf. NMA'98, Sofia (Bulgarie) (Iliev, O.P.; Kaschiev, M.S.; Margenov, S.D.; Sendov, Bl.H.; Vassilevski, P.S., eds.), World Sci. Publications (1999), pp. 621-629

[3] Angot, Ph.; Gallouët, Th.; Herbin, R. Convergence of finite volume methods on general meshes for non smooth solution of elliptic problems with cracks (Vilsmeier, R.; Benkhaldoun, F.; Hänel, D., eds.), Finite Volumes for Complex Applications II, Hermès, 1999, pp. 215-222

[4] Ph. Angot, A model of fracture for elliptic problems with flux and solution jumps, Preprint L.A.T.P., UMR CNRS 6632 http://www.cmi.univ-mrs.fr, May 2003

[5] Ph. Angot, Th. Gallouët, R. Herbin, in preparation

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[7] Eymard, R.; Gallouët, Th.; Herbin, R. Finite volume methods (Ciarlet, P.G.; Lions, J.-L., eds.), Handbook of Numerical Analysis, Vol. VII, North-Holland, 2000, pp. 713-1020

[8] Faille, I.; Flauraud, E.; Nataf, F.; Pégaz-Fiornet, S.; Schneider, F.; Willien, F. A new fault model in geological basin modelling. Application of finite volume scheme and domain decomposition methods (Herbin, R.; Kröner, D., eds.), Finite Volumes for Complex Applications III, Hermes Penton Sci. (HPS), 2002, pp. 543-550

[9] Dynamics of Fluids in Fractured Rock (Faybishenko, B.; Witherspoon, P.A.; Benson, S.M., eds.), Geophysical Monograph Series, 122, American Geophysical Union, Washington, DC, 2000

[10] Jaffré, J.; Martin, V.; Roberts, J.E. Generalized cell-centered finite volume methods for flow in porous media with faults (Herbin, R.; Kröner, D., eds.), Finite Volumes for Complex Applications III, Hermes Penton Sci. (HPS), 2002, pp. 357-364

Cited by Sources:

An extended version of this paper with some more details can be found in Angot (Preprint L.A.T.P.).