Error estimate for a finite volume scheme in a geometrical multi-scale domain. (English) Zbl 1317.65225
The author considers a finite volume scheme, introduced in a previous paper [G. Panasenko and M.-C. Viallon, Math. Methods Appl. Sci. 36, No. 14, 1892–1917 (2013; Zbl 1273.65165)], to solve an elliptic linear partial differential equation appearing 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. For this reason, the considered structure is called a geometrical multi-scale domain. A discrete Poincaré inequality in terms of a specific \(H^1\) norm defined on this geometrical multi-scale 1D-2D domain is established. The stated Poincaré inequality 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. An \(L^2\) error estimate between the solution of the equation and its numerical finite volume approximation is derived.
Reviewer: Abdallah Bradji (Annaba)
MSC:
| 65N15 | Error bounds for boundary value problems involving PDEs |
| 65N08 | Finite volume methods for boundary value problems involving PDEs |
| 74S10 | Finite volume methods applied to problems in solid mechanics |
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
| 35J25 | Boundary value problems for second-order elliptic equations |
| 74K10 | Rods (beams, columns, shafts, arches, rings, etc.) |
Keywords:
rod structure; elliptic linear partial differential equation; finite volume scheme; multi-scale domain; discrete Poincaré inequality; error estimateCitations:
Zbl 1273.65165References:
| [1] | 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. · Zbl 1019.65086 · doi:10.1007/s002110100336 |
| [2] | 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 Fluids56 (2008) 1123-1129. · Zbl 1384.65051 · doi:10.1002/fld.1680 |
| [3] | 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. · Zbl 1111.65101 · doi:10.1002/num.20170 |
| [4] | 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. · Zbl 1228.65178 · doi:10.1142/S0218202511005064 |
| [5] | 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). |
| [6] | 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. · Zbl 1173.76430 · doi:10.1016/j.cma.2007.05.008 |
| [7] | 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. · Zbl 1228.76203 · doi:10.1016/j.cma.2010.12.006 |
| [8] | 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:10.1016/j.jbiomech.2009.01.034 |
| [9] | 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 1193.92027 |
| [10] | 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. · Zbl 1211.35017 · doi:10.1142/S0218202510004817 |
| [11] | 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. · Zbl 1065.65137 · doi:10.1093/imanum/24.3.465 |
| [12] | 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. · Zbl 1211.65144 · doi:10.1093/imanum/drp009 |
| [13] | Y. Coudière, J.-P. Vila and P. Villedieu, Convergence rate of a finite volume scheme for a two-dimensional convection-diffusion problem. ESAIM: M2AN33 (1999) 493-516. · Zbl 0937.65116 · doi:10.1051/m2an:1999149 |
| [14] | Y. Coudière, T. Gallouët and R. Herbin, Discrete Sobolev Inequalities and L^{p} error estimates for finite volume solutions of convection diffusion equations. ESAIM: M2AN35 (2001) 767-778. · Zbl 0990.65122 · doi:10.1051/m2an:2001135 |
| [15] | K. Domelevo and P. Omnes, A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids. ESAIM: M2AN39 (2005) 1203-1249. · Zbl 1086.65108 · doi:10.1051/m2an:2005047 |
| [16] | 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. · Zbl 1329.76227 |
| [17] | 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. · Zbl 1058.65127 · doi:10.1137/S0036142902405205 |
| [18] | 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. · Zbl 1202.65144 · doi:10.1093/imanum/drn084 |
| [19] | R. Eymard, T. Gallouët and R. Herbin, Finite Volume Methods. Handb. Numer. Anal. Edited by P.G. Ciarlet and J.L. Lions (2000). · Zbl 0981.65095 |
| [20] | F. Filbet, A finite volume scheme for the Patlak-Keller-Segel chemotaxis model. Numer. Math.104 (2006) 457-488. · Zbl 1098.92006 · doi:10.1007/s00211-006-0024-3 |
| [21] | F. Fontvieille, G.P. Panasenko and J. Pousin, FEM implementation for the asymptotic partial decomposition. Appl. Anal. Int. J.86 (2007) 519-536. · Zbl 1115.65114 · doi:10.1080/00036810601106545 |
| [22] | 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. · Zbl 1067.76624 · doi:10.1007/s007910050030 |
| [23] | 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. · Zbl 1007.74035 · doi:10.1016/S0045-7825(01)00302-4 |
| [24] | L. Formaggia, A. Quarteroni and A. Veneziani, Cardiovascular Mathematics, Series: Model. Simul. Appl., vol. 1. Springer (2009). |
| [25] | 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. · Zbl 0986.65099 · doi:10.1137/S0036142999351388 |
| [26] | A. Glitzky and J.A. Griepentrog, Discrete Sobolev-Poincaré Inequalities for Voronoi Finite Volume Approximations. SIAM J. Numer. Anal.48 (2010) 372-391. · Zbl 1209.46015 · doi:10.1137/09076502X |
| [27] | P. Grisvard, Elliptic Problems in Non Smooth Domains. Pitman (1985). · Zbl 0695.35060 |
| [28] | J.M. Hérard and O. Hurisse, Coupling two and one-dimensional unsteady Euler equations through a thin interface. Comput. Fluids36 (2007) 651-666. · Zbl 1177.76228 · doi:10.1016/j.compfluid.2006.03.007 |
| [29] | 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. · Zbl 0822.65085 · doi:10.1002/num.1690110205 |
| [30] | 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. · Zbl 0863.76016 · doi:10.1002/(SICI)1097-0363(19960315)22:5<325::AID-FLD307>3.0.CO;2-Y |
| [31] | 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. · Zbl 1288.65151 |
| [32] | 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. · Zbl 1183.76838 |
| [33] | 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. · Zbl 1300.76011 · doi:10.1137/100809301 |
| [34] | 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. · Zbl 1310.92017 · doi:10.1137/120867408 |
| [35] | G.P. Panasenko, Method of asymptotic partial decomposition of domain. Math. Models Methods Appl. Sci.8 (1998) 139-156. · Zbl 0940.35026 · doi:10.1142/S021820259800007X |
| [36] | 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. · Zbl 1273.65165 · doi:10.1002/mma.2735 |
| [37] | 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. · Zbl 1154.35310 · doi:10.1080/00036810802282533 |
| [38] | 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. · Zbl 1168.76394 · doi:10.1007/s10665-009-9281-3 |
| [39] | 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). |
| [40] | 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. · Zbl 1088.65100 · doi:10.1137/S0036142903434059 |
| [41] | 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. · Zbl 1178.76395 · doi:10.1016/j.cma.2005.07.014 |
| [42] | 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. I351 (2013) 47-51. · Zbl 1261.65110 · doi:10.1016/j.crma.2013.01.011 |
| [43] | 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. · Zbl 1089.65124 · doi:10.1080/01630560500444533 |
| [44] | 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. |
| [45] | 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: M2AN47 (2013) 961-985. · Zbl 1402.92143 · doi:10.1051/m2an/2012053 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
