Error estimate for a finite volume scheme in a geometrical multi-scale domain

Viallon, Marie-Claude

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 2, pp. 529-550.

We study a finite volume scheme, introduced in a previous paper [G.P. Panasenko and M.-C. Viallon, Math. Meth. Appl. Sci. 36 (2013) 1892–1917], to solve an elliptic linear partial differential equation in a rod structure. The rod-structure is two-dimensional (2D) and consists of a central node and several outgoing branches. The branches are assumed to be one-dimensional (1D). So the domain is partially 1D, and partially 2D. We call such a structure a geometrical multi-scale domain. We establish a discrete Poincaré inequality in terms of a specific $H^1$ norm defined on this geometrical multi-scale 1D-2D domain, that is valid for functions that satisfy a Dirichlet condition on the boundary of the 1D part of the domain and a Neumann condition on the boundary of the 2D part of the domain. We derive an $L^2$ error estimate between the solution of the equation and its numerical finite volume approximation.

Classification : 35J25, 74S10, 65N12, 65N15, 65N08
Mots clés : Finite volume scheme, elliptic problem, discrete Poincaré inequality, error estimate, multi-scale domain

Y. Achdou, C. Japhet, Y. Maday and F. Nataf, A new cement to glue non-conforming grids with Robin interface conditions: the finite volume case. Numer. Math. 92 (2002) 593–620. | DOI | MR | Zbl

A. Ambroso, C. Chalons, F. Coquel, E. Godlewski, F. Lagoutière, P.A. Raviart and N. Seguin, Relaxation methods and coupling procedures. Int. J. Numer. Methods Fluids 56 (2008) 1123–1129. | DOI | MR | Zbl

B. Andreianov, F. Boyer and F. Hubert, Discrete duality finite volume schemes for Leray-Lions-type elliptic problems on general 2D meshes. Numer. Method Partial Differ. Eq. 23 (2007) 145–195. | DOI | MR | Zbl

B. Andreianov, M. Bendahmane and R. Ruiz Baier, Analysis of a finite volume method for a cross-diffusion model in population dynamics. Math. Meth. Appl. Sci. 21 (2011) 307–344. | DOI | MR | Zbl

M. Bessemoulin-Chatard, C. Chainais-Hillairet and F. Filbet, On discrete functional inequalities for some finite volume schemes. To appear in IMA J. Numer. Anal. (2014). | MR

P.J. Blanco, R.A. Feijóo and S.A. Urquiza, A unified variational approach for coupling 3D-1D models and its blood flow applications. Comput. Methods Appl. Mech. Eng. 196 (2007) 4391–4410. | DOI | MR | Zbl

P.J. Blanco, J.S. Leiva, R.A. Feijóo and G.S. Buscaglia, Black-box decomposition approach for computational hemodynamics: One-dimensional models. Comput. Methods Appl. Mech. Eng. 200 (2011) 1389–1405. | DOI | MR | Zbl

P.J. Blanco, M.R. Pivello, S.A. Urquiza and R.A. Feijóo, On the potentialities of 3D-1D coupled models in hemodynamics simulations. J. Biomech. 42 (2009) 919–930. | DOI

P.J. Blanco, S.A. Urquiza and R.A. Feijóo, Assessing the influence of heart rate in local hemodynamics through coupled 3D-1D-0D models. Int. J. Numer. Methods Biomed. Eng. 26 (2010) 890–903. | Zbl

B. Boutin, C. Chalons and P.A. Raviart, Existence result for the coupling problem of two scalar conservation laws with Riemann initial data. Math. Models Methods Appl. Sci. 20 (2010) 1859–1898. | DOI | MR | Zbl

R. Cautrés, R. Herbin and F. Hubert, The Lions domain decomposition algorithm on non matching cell-centred finite volume meshes. IMA J. Numer. Anal. 24 (2004) 465–490. | DOI | MR | Zbl

C. Chainais-Hillairet and J. Droniou, Finite volume schemes for non-coercive elliptic problems with Neumann boundary conditions. IMA J. Numer. Anal. 31 (2011) 61–85. | DOI | MR | Zbl

Y. Coudière, J.-P. Vila and P. Villedieu, Convergence rate of a finite volume scheme for a two-dimensional convection-diffusion problem. ESAIM: M2AN 33 (1999) 493–516. | DOI | Numdam | MR | Zbl

Y. Coudière, T. Gallouët and R. Herbin, Discrete Sobolev Inequalities and $L^p$ error estimates for finite volume solutions of convection diffusion equations. ESAIM: M2AN 35 (2001) 767–778. | DOI | Numdam | MR | Zbl

K. Domelevo and P. Omnes, A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids. ESAIM: M2AN 39 (2005) 1203–1249. | DOI | Numdam | MR | Zbl

M. Deininger, J. Jung, R. Skoda, P. Helluy and C.-D. Munz, Evaluation of interface models for 3D-1D coupling of compressible Euler methods for the application on cavitating flows. CEMRACS’11: Multiscale coupling of complex models in scientific computing. ESAIM Proceedings. EDP Sciences Les Ulis 38 (2012) 298–318. | MR | Zbl

J. Droniou, T. Gallouët and R. Herbin, A finite volume scheme for a noncoercive elliptic equation with measure data. SIAM J. Numer. Anal. 41 (2003) 1997–2031. | DOI | MR | Zbl

R. Eymard, T. Gallouët and R. Herbin, Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes SUSHI: a scheme using stabilization and hybrid interfaces. IMA J. Numer. Anal. 30 (2010) 1009–1043. | DOI | MR | Zbl

R. Eymard, T. Gallouët and R. Herbin, Finite Volume Methods. Handb. Numer. Anal. Edited by P.G. Ciarlet and J.L. Lions (2000). | MR | Zbl

F. Filbet, A finite volume scheme for the Patlak-Keller-Segel chemotaxis model. Numer. Math. 104 (2006) 457–488. | DOI | MR | Zbl

F. Fontvieille, G.P. Panasenko and J. Pousin, FEM implementation for the asymptotic partial decomposition. Appl. Anal. Int. J. 86 (2007) 519–536. | DOI | MR | Zbl

L. Formaggia, F. Nobile, A. Quarteroni and A. Veneziani, Multiscale modelling of the circulatory system: a preliminary analysis. Comput. Visual. Sci. 2 (1999) 75–83. | DOI | Zbl

L. Formaggia, J.F. Gerbeau, F. Nobile and A. Quarteroni, On the coupling of 3D and 1D Navier–Stokes equations for flow problems in compliant vessels. Comput. Methods Appl. Mech. Eng. 191 (2001) 561–582. | DOI | MR | Zbl

L. Formaggia, A. Quarteroni and A. Veneziani, Cardiovascular Mathematics, Series: Model. Simul. Appl., vol. 1. Springer (2009). | MR | Zbl

T. Gallouët, R. Herbin and M.H. Vignal, Error estimates on the approximate finite volume solution of convection diffusion equations with general boundary conditions. SIAM J. Numer. Anal. 37 (2000) 1935–1972. | DOI | MR | Zbl

A. Glitzky and J.A. Griepentrog, Discrete Sobolev–Poincaré Inequalities for Voronoi Finite Volume Approximations. SIAM J. Numer. Anal. 48 (2010) 372–391. | DOI | MR | Zbl

P. Grisvard, Elliptic Problems in Non Smooth Domains. Pitman (1985). | MR | Zbl

J.M. Hérard and O. Hurisse, Coupling two and one-dimensional unsteady Euler equations through a thin interface. Comput. Fluids 36 (2007) 651–666. | DOI | Zbl

R. Herbin, An error estimate for a finite volume scheme for a diffusion-convection problem on a triangular mesh. Numer. Method Partial Differ. Eq. 11 (1995) 165–173. | DOI | MR | Zbl

J. Heywood, R. Rannacher and S. Turek, Artificial boundaries and flux and pressure conditions for the incompressible Navier–Stokes equations. Int. J. Num. Meth. Fl. 22 (1996) 325–352. | DOI | MR | Zbl

A.H. Le and P. Omnes, Discrete Poincaré inequalities for arbitrary meshes in the discrete duality finite volume context. Electronic Trans. Numer. Anal. 40 (2013) 94–119. | MR | Zbl

J.S. Leiva, P.J. Blanco and G.S. Buscaglia, Iterative strong coupling of dimensionally heterogeneous models. Int. J. Numer. Methods Eng. 81 (2010) 1558–1580. | MR | Zbl

J.S. Leiva, P.J. Blanco and G.S. Buscaglia, Partitioned analysis for dimensionally-heterogeneous hydraulic networks. SIAM Multiscale Model. Simul. 9 (2011) 872–903. | DOI | MR | Zbl

A.C.I. Malossi, P.J. Blanco, P. Crosetto, S. Deparis and A. Quarteroni, Implicit coupling of one-dimensional and three-dimensional blood flow models with compliant vessels. Multiscale Model. Simul. 11 (2013) 474–506. | DOI | MR | Zbl

G.P. Panasenko, Method of asymptotic partial decomposition of domain. Math. Models Methods Appl. Sci. 8 (1998) 139–156. | DOI | MR | Zbl

G.P. Panasenko and M.-C. Viallon, Error estimate in a finite volume approximation of the partial asymptotic domain decomposition. Math. Meth. Appl. Sci. 36 (2013) 1892–1917. | DOI | MR | Zbl

G.P. Panasenko and M.-C. Viallon, The finite volume implementation of the partial asymptotic domain decomposition. Appl. Anal. Int. J. 87 (2008) 1397–1424. | DOI | MR | Zbl

T. Passerini, M. De Luca, L. Formaggia and A. Quarteroni, A 3D/1D geometrical multiscale model of cerebral vasculature. J. Eng. Math. 64 (2009) 319–330. | DOI | MR | Zbl

A. Quarteroni and L. Formaggia, Mathematical Modelling and Numerical Simulation of the Cardiovascular System. Modelling of Living Systems. Edited by N. Ayache. Handb. Numer. Anal. Series (2002). | MR

L. Saas, I. Faille, F. Nataf and F. Willien, Finite volume methods for domain decomposition on non matching grids with arbitrary interface conditions. SIAM J. Numer. Anal. 43 (2005) 860–890. | DOI | MR | Zbl

S.A. Urquiza, P.J. Blanco, M.J. Vénere and R.A. Feijóo, Multidimensional modelling for the carotid artery blood flow. Comput. Methods Appl. Mech. Eng. 195 (2006) 4002–4017. | DOI | MR | Zbl

M.-C. Viallon, Error estimate for a 1D-2D finite volume scheme. Comparison with a standard scheme on a 2D non-admissible mesh. C. R. Acad. Sci. Paris, Ser. I 351 (2013) 47–51. | DOI | MR | Zbl

M. Vohralik, On the discrete Poincaré-Friedrichs inequalities for nonconforming approximations of the sobolev space $H^1$. Numer. Funct. Anal. Optim. 26 (2005) 925–952. | DOI | MR | Zbl

M. Vohralik, Numerical methods for nonlinear elliptic and parabolic equations. Application to flow problems in porous and fractured media. Ph.D. thesis, Université de Paris-Sud and Czech Technical University in Prague.

S.M. Watanabe, P.J. Blanco and R.A. Feijóo, Mathematical model of blood flow in an anatomically detailed arterial network of the arm. ESAIM: M2AN 47 (2013) 961–985. | DOI | Numdam | MR | Zbl