Voir la notice de l'article provenant de la source Numdam
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 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 error estimate between the solution of the equation and its numerical finite volume approximation.
Viallon, Marie-Claude 1
@article{M2AN_2015__49_2_529_0,
author = {Viallon, Marie-Claude},
title = {Error estimate for a finite volume scheme in a geometrical multi-scale domain},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {529--550},
publisher = {EDP-Sciences},
volume = {49},
number = {2},
year = {2015},
doi = {10.1051/m2an/2014042},
mrnumber = {3342216},
zbl = {1317.65225},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an/2014042/}
}
TY - JOUR AU - Viallon, Marie-Claude TI - Error estimate for a finite volume scheme in a geometrical multi-scale domain JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2015 SP - 529 EP - 550 VL - 49 IS - 2 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/m2an/2014042/ DO - 10.1051/m2an/2014042 LA - en ID - M2AN_2015__49_2_529_0 ER -
%0 Journal Article %A Viallon, Marie-Claude %T Error estimate for a finite volume scheme in a geometrical multi-scale domain %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2015 %P 529-550 %V 49 %N 2 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/m2an/2014042/ %R 10.1051/m2an/2014042 %G en %F M2AN_2015__49_2_529_0
Viallon, Marie-Claude. Error estimate for a finite volume scheme in a geometrical multi-scale domain. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 2, pp. 529-550. doi : 10.1051/m2an/2014042. http://geodesic.mathdoc.fr/articles/10.1051/m2an/2014042/
, , and , A new cement to glue non-conforming grids with Robin interface conditions: the finite volume case. Numer. Math. 92 (2002) 593–620. | MR | Zbl | DOI
, , , , , and , Relaxation methods and coupling procedures. Int. J. Numer. Methods Fluids 56 (2008) 1123–1129. | MR | Zbl | DOI
, and , Discrete duality finite volume schemes for Leray-Lions-type elliptic problems on general 2D meshes. Numer. Method Partial Differ. Eq. 23 (2007) 145–195. | MR | Zbl | DOI
, and Baier, Analysis of a finite volume method for a cross-diffusion model in population dynamics. Math. Meth. Appl. Sci. 21 (2011) 307–344. | MR | Zbl | DOI
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
, and , A unified variational approach for coupling 3D-1D models and its blood flow applications. Comput. Methods Appl. Mech. Eng. 196 (2007) 4391–4410. | MR | Zbl | DOI
, , and , Black-box decomposition approach for computational hemodynamics: One-dimensional models. Comput. Methods Appl. Mech. Eng. 200 (2011) 1389–1405. | MR | Zbl | DOI
, , and , On the potentialities of 3D-1D coupled models in hemodynamics simulations. J. Biomech. 42 (2009) 919–930. | DOI
, and , 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
, and , Existence result for the coupling problem of two scalar conservation laws with Riemann initial data. Math. Models Methods Appl. Sci. 20 (2010) 1859–1898. | MR | Zbl | DOI
, and , The Lions domain decomposition algorithm on non matching cell-centred finite volume meshes. IMA J. Numer. Anal. 24 (2004) 465–490. | MR | Zbl | DOI
and , Finite volume schemes for non-coercive elliptic problems with Neumann boundary conditions. IMA J. Numer. Anal. 31 (2011) 61–85. | MR | Zbl | DOI
, and , Convergence rate of a finite volume scheme for a two-dimensional convection-diffusion problem. ESAIM: M2AN 33 (1999) 493–516. | MR | Zbl | mathdoc-id | DOI
, and , Discrete Sobolev Inequalities and error estimates for finite volume solutions of convection diffusion equations. ESAIM: M2AN 35 (2001) 767–778. | MR | Zbl | mathdoc-id | DOI
and , A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids. ESAIM: M2AN 39 (2005) 1203–1249. | MR | Zbl | mathdoc-id | DOI
, , , and , 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
, and , A finite volume scheme for a noncoercive elliptic equation with measure data. SIAM J. Numer. Anal. 41 (2003) 1997–2031. | MR | Zbl | DOI
, and , 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. | MR | Zbl | DOI
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
, A finite volume scheme for the Patlak-Keller-Segel chemotaxis model. Numer. Math. 104 (2006) 457–488. | MR | Zbl | DOI
, and , FEM implementation for the asymptotic partial decomposition. Appl. Anal. Int. J. 86 (2007) 519–536. | MR | Zbl | DOI
, , and , Multiscale modelling of the circulatory system: a preliminary analysis. Comput. Visual. Sci. 2 (1999) 75–83. | Zbl | DOI
, , and , 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. | MR | Zbl | DOI
L. Formaggia, A. Quarteroni and A. Veneziani, Cardiovascular Mathematics, Series: Model. Simul. Appl., vol. 1. Springer (2009). | MR | Zbl
, and , Error estimates on the approximate finite volume solution of convection diffusion equations with general boundary conditions. SIAM J. Numer. Anal. 37 (2000) 1935–1972. | MR | Zbl | DOI
and , Discrete Sobolev–Poincaré Inequalities for Voronoi Finite Volume Approximations. SIAM J. Numer. Anal. 48 (2010) 372–391. | MR | Zbl | DOI
P. Grisvard, Elliptic Problems in Non Smooth Domains. Pitman (1985). | MR | Zbl
and , Coupling two and one-dimensional unsteady Euler equations through a thin interface. Comput. Fluids 36 (2007) 651–666. | Zbl | DOI
, 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. | MR | Zbl | DOI
, and , Artificial boundaries and flux and pressure conditions for the incompressible Navier–Stokes equations. Int. J. Num. Meth. Fl. 22 (1996) 325–352. | MR | Zbl | 3.0.CO;2-Y class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI
and , Discrete Poincaré inequalities for arbitrary meshes in the discrete duality finite volume context. Electronic Trans. Numer. Anal. 40 (2013) 94–119. | MR | Zbl
, and , Iterative strong coupling of dimensionally heterogeneous models. Int. J. Numer. Methods Eng. 81 (2010) 1558–1580. | MR | Zbl
, and , Partitioned analysis for dimensionally-heterogeneous hydraulic networks. SIAM Multiscale Model. Simul. 9 (2011) 872–903. | MR | Zbl | DOI
, , , and , Implicit coupling of one-dimensional and three-dimensional blood flow models with compliant vessels. Multiscale Model. Simul. 11 (2013) 474–506. | MR | Zbl | DOI
, Method of asymptotic partial decomposition of domain. Math. Models Methods Appl. Sci. 8 (1998) 139–156. | MR | Zbl | DOI
and , Error estimate in a finite volume approximation of the partial asymptotic domain decomposition. Math. Meth. Appl. Sci. 36 (2013) 1892–1917. | MR | Zbl | DOI
and , The finite volume implementation of the partial asymptotic domain decomposition. Appl. Anal. Int. J. 87 (2008) 1397–1424. | MR | Zbl | DOI
, , and , A 3D/1D geometrical multiscale model of cerebral vasculature. J. Eng. Math. 64 (2009) 319–330. | MR | Zbl | DOI
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
, , and , Finite volume methods for domain decomposition on non matching grids with arbitrary interface conditions. SIAM J. Numer. Anal. 43 (2005) 860–890. | MR | Zbl | DOI
, , and , Multidimensional modelling for the carotid artery blood flow. Comput. Methods Appl. Mech. Eng. 195 (2006) 4002–4017. | MR | Zbl | DOI
, 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. | MR | Zbl | DOI
, On the discrete Poincaré-Friedrichs inequalities for nonconforming approximations of the sobolev space . Numer. Funct. Anal. Optim. 26 (2005) 925–952. | MR | Zbl | DOI
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.
, and , Mathematical model of blood flow in an anatomically detailed arterial network of the arm. ESAIM: M2AN 47 (2013) 961–985. | MR | Zbl | mathdoc-id | DOI
Cité par Sources :