Convergence rate of a finite volume scheme for the linear convection-diffusion equation on locally refined meshes
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 34 (2000) no. 6, p. 1123-1149
@article{M2AN_2000__34_6_1123_0,
     author = {Coudi\`ere, Yves and Villedieu, Philippe},
     title = {Convergence rate of a finite volume scheme for the linear convection-diffusion equation on locally refined meshes},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {Dunod},
     volume = {34},
     number = {6},
     year = {2000},
     pages = {1123-1149},
     zbl = {0972.65081},
     mrnumber = {1812729},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2000__34_6_1123_0}
}
Coudière, Yves; Villedieu, Philippe. Convergence rate of a finite volume scheme for the linear convection-diffusion equation on locally refined meshes. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 34 (2000) no. 6, pp. 1123-1149. http://www.numdam.org/item/M2AN_2000__34_6_1123_0/

[1] R.E. Bank and D.J. Rose, Some error estimates for the box method. SIAM J. Numer. Anal. 24 (1987) 777 787. | MR 899703 | Zbl 0634.65105

[2] J. Baranger, J.F. Maitre and F. Oudin, Connection between finite volume and mixed finite element methods. RAIRO Modél. Math. Anal. Numér. 30 (1996) 445-465. | Numdam | MR 1399499 | Zbl 0857.65116

[3] M.J. Berger and P. Collela, Local adaptative mesh refinement for shock hydrodynamics. J. Comput. Phys. 82 (1989) 64-84. | Zbl 0665.76070

[4] Z. Cai, On the finite volume element method. Numer. Math. 58 (1991) 713-735. | MR 1090257 | Zbl 0731.65093

[5] Z. Cai, J. Mandel and S. Mccormick, The finite volume element method for diffusion equations on general triangulations. SIAM J. Numer. Anal. 28 (1991) 392-402. | MR 1087511 | Zbl 0729.65086

[6] Z. Cai and S. Mccormick, On the accuracy of the finite volume element method for diffusion equations on composite grids. SIAM J. Numer. Anal 27 (1990) 636-655. | MR 1041256 | Zbl 0707.65073

[7] W.J. Coirier, An Adaptatively-Refined, Cartesian, Cell-based Scheme for the Euler and Navier-Stokes Equations. Ph.D. thesis, Michigan Univ., NASA Lewis Research Center (1994).

[8] W.J. Coirier and K.G. Powell, A Cartesian, cell-based approach for adaptative-refined solutions of the Euler and Navier-Stokes equations. AIAA (1995).

[9] Y. Coudière, Analyse de schémas volumes finis sur maillages non structurés pour des problèmes linéaires hyperboliques et elliptiques. Ph.D. thesis, Université Paul Sabatier (1999).

[10] Y. Coudière, T. Gallouët and R. Herbin, Discrete sobolev inequalities and lp error estimates for approximate finite volume solutions of convection diffusion equation. Preprint of LATP, University of Marseille 1, 98-13 (1998).

[11] Y. Coudière, J.P. Vila and P. Villedieu, Convergence rate of a finite volume scheme for a two dimensionnal diffusion convection problem. ESAIM: M2AN 33 (1999) 493-516. | Numdam | MR 1713235 | Zbl 0937.65116

[12] B. Courbet and J.P. Croisille, Finite volume box schemes on triangular meshes. RAIRO Modél. Math. Anal. Numér. 32 (1998) 631-649. | Numdam | MR 1643473 | Zbl 0920.65065

[13] M. Dauge, Elliptic Boundary Value Problems in Corner Domains. Lect. Notes Math., Springer-Verlag, Berlin (1988). | MR 961439 | Zbl 0668.35001

[14] R.E. Ewing, R.D. Lazarov and P.S. Vassilevski, Local refinement techniques for elliptic problems on cell-centered grids. I. Error analysis. Math. Comp. 56 (1991) 437-461. | MR 1066831 | Zbl 0724.65093

[15] R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, in Handbook of Numerical Analysis, P.G. Ciarlet and J.L. Lions Eds. (to appear). Prépublication No 97-19 du LATP, UMR 6632, Marseille (1997). | MR 1804748 | Zbl 0981.65095

[16] P.A. Forsyth and P.H. Sammon, Quadratic convergence for cell-centered grids. Appl Numer. Math. 4 (1988) 377-394. | MR 948505 | Zbl 0651.65086

[17] B. Heinrich, Finite Difference Methods on Irregular Networks. Internat. Ser. Numer. Anal. 82, Birkhaüser, Verlag Basel (1987). | MR 875416 | Zbl 0623.65096

[18] R. Herbin, An error estimate for a finite volume scheme for a diffusion-convection problem on a triangular mesh. Numer. Methods Partial Differential Equations 11 (1994) 165-173. | MR 1316144 | Zbl 0822.65085

[19] F. Jacon and D. Knight, A Navier-Stokes algorithm for turbulent flows using an unstructured grid and flux difference splitting. AIAA (1994).

[20] H. Jianguo and X. Shitong, On the finite volume element method for general self-adjoint elliptic problem. SIAM J. Numer. Anal. 35 (1998) 1762-1774. | MR 1640017 | Zbl 0913.65097

[21] P. Lesaint, Sur la résolution des systèmes hyperboliques du premier ordre par des méthodes d'éléments finis. Technical report, CEA (1976).

[22] T.A. Manteuffel and A.B. White, The numerical solution of second-order boundary values problems on nonuniform meshes. Math. Comp. 47 (1986) 511-535. | MR 856700 | Zbl 0635.65092

[23] K. Mer, Variational analysis of a mixed finite element finite volume scheme on general triangulations. Technical Report 2213, INRIA, Sophia Antipolis (1994).

[24] I.D. Mishev, Finite volume methods on voronoï meshes. Numer. Methods Partial Differential Equations 14 (1998) 193-212. | MR 1605410 | Zbl 0903.65083

[25] K.W. Morton and E. Süli, Finite volume methods and their analysis. IMA J. Numer. Anal. 11 (1991) 241-260. | MR 1105229 | Zbl 0729.65087

[26] E. Süli, Convergence of finite volume schemes for Poisson's equation on nonuniform meshes. SIAM J. Numer. Anal. 28 (1991) 1419-1430. | MR 1119276 | Zbl 0802.65104

[27] J.-M. Thomas and D. Trujillo. Analysis of finite volumes methods. Technical Report 95/19, CNRS, URA 1204 (1995).

[28] J.-M. Thomas and D. Trujillo, Convergence of finite volumes methods. Technical Report 95/20, CNRS, URA 1204 (1995).

[29] R. Vanselow and H.P. Scheffler, Convergence analysis of a finite volume method via a new nonconforming finite element method. Numer. Methods Partial Differential Equations 14 (1998) 213-231. | MR 1605414 | Zbl 0903.65084

[30] P.S. Vassilevski, S.I. Petrova and R.D. Lazarov. Finite difference schemes on triangular cell-centered grids with local refinement. SIAM J. Sci. Stat. Comput. 13 (1992) 1287-1313. | MR 1185647 | Zbl 0813.65115

[31] A. Weiser and M.F. Wheeler, On convergence of block-centered finite differences for elliptic problems. SIAM J. Numer. Anal. 25 (1988) 351-375. | MR 933730 | Zbl 0644.65062