Preserving monotony of combined edge finite volume-finite element scheme for a bone healing model on general mesh

In this article, we propose and analyze a combined finite volume-finite element scheme for a bone healing model. This choice of discretization allows to take into account anisotropic diffusions without imposing any restrictions on the mesh. Moreover, following the work of Cances etźal. (2013), we define a nonlinear correction of the diffusive terms to obtain a monotone scheme. We provide, under a numerical assumption, a complete convergence analysis of this corrected scheme, and present some numerical experiments which show its good behavior.

[1]  P. Raviart,et al.  Conforming and nonconforming finite element methods for solving the stationary Stokes equations I , 1973 .

[2]  M. V. D. van der Meulen,et al.  A mathematical framework to study the effects of growth factor influences on fracture healing. , 2001, Journal of theoretical biology.

[3]  Mazen Saad,et al.  Analysis of a finite volume method for a bone growth system in vivo , 2013, Comput. Math. Appl..

[4]  Martin Vohralík,et al.  A combined finite volume–nonconforming/mixed-hybrid finite element scheme for degenerate parabolic problems , 2006, Numerische Mathematik.

[5]  Mazen Saad,et al.  A combined finite volume–nonconforming finite element scheme for compressible two phase flow in porous media , 2013, Numerische Mathematik.

[6]  M. Vohralík.  On the Discrete Poincaré–Friedrichs Inequalities for Nonconforming Approximations of the Sobolev Space H 1 , 2005 .

[7]  Clément Cancès,et al.  Monotone corrections for generic cell-centered finite volume approximations of anisotropic diffusion equations , 2013, Numerische Mathematik.

[8]  R. Temam.  Navier-Stokes Equations , 1977 .

[9]  Todd Arbogast,et al.  A Nonlinear Mixed Finite Eelement Method for a Degenerate Parabolic Equation Arising in Flow in Porous Media , 1996 .

[10]  Martin Vohralík,et al.  Numerical methods for nonlinear elliptic and parabolic equations. Application to flow problems in porous and fractured media. (Méthodes numériques pour des équations elliptiques et paraboliques non linéaires. Application à des problèmes d'écoulement en milieux poreux et fracturés) , 2004 .

[11]  R. Eymard,et al.  Discretisation of heterogeneous and anisotropic diffusion problems on general non-conforming meshes. SUSHI: a scheme using stabilisation and hybrid interfaces , 2008, 0801.1430.

[12]  Thierry Gallouët,et al.  Convergence of finite volume schemes for semilinear convection diffusion equations , 1999, Numerische Mathematik.

[13]  Christophe Le Potier.  Correction non linéaire et principe du maximum pour la discrétisation d'opérateurs de diffusion avec des schémas volumes finis centrés sur les mailles , 2010 .

[14]  John W. Barrett,et al.  Finite Element Approximation of the Transport of Reactive Solutes in Porous Media. Part 1: Error Estimates for Nonequilibrium Adsorption Processes , 1997 .

[15]  M. Lukáčová-Medvid'ová,et al.  Combined finite element-finite volume solution of compressible flow , 1995 .

[16]  Mazen Saad,et al.  Monotone combined finite volume-finite element scheme for a bone healing model , 2014 .

[17]  Alexandre Ern,et al.  Discrete maximum principle for Galerkin approximations of the Laplace operator on arbitrary meshes , 2004 .

[18]  Alexandre Uzureau.  Modélisations et calculs pour la cicatrisation osseuse. Application à la modélisation d'un bioréacteur. , 2012 .

[19]  Long Chen.  FINITE VOLUME METHODS , 2011 .

[20]  Georges Chamoun,et al.  Monotone combined edge finite volume–finite element scheme for Anisotropic Keller–Segel model , 2014 .

[21]  H. Brezis.  Functional Analysis, Sobolev Spaces and Partial Differential Equations , 2010 .