×

zbMATH — the first resource for mathematics

Convergence rate of a finite volume scheme for the linear convection-diffusion equation on locally refined meshes. (English) Zbl 0972.65081
The numerical solution of the linear two-dimensional convection-diffusion equation, with mixed Dirichlet and Neumann boundary conditions, is considered. For the approximation the finite volume method on Cartesian meshes refined using an automatic technique is proposed. This technique leads to meshes with hanging nodes. An analysis through a discrete variational approach in a discrete \(H^1\) finite volume space is used. The convergence of this scheme in a discrete \(H^1\) norm, with an error estimate of order \(O(h)\), where \(h\) is the size of the mesh is proved.

MSC:
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
65N15 Error bounds for boundary value problems involving PDEs
PDF BibTeX XML Cite
Full Text: DOI Link EuDML
References:
[1] R.E. Bank and D.J. Rose, Some error estimates for the box method. SIAM J. Numer. Anal.24 (1987) 777-787. · Zbl 0634.65105 · doi:10.1137/0724050
[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. · Zbl 0857.65116 · eudml:193811
[3] M.J. Berger and P. Collela, Local adaptative mesh refinement for shock hydrodynamics. J. Comput. Phys.82 (1989) 64-84. Zbl0665.76070 · Zbl 0665.76070 · doi:10.1016/0021-9991(89)90035-1
[4] Z. Cai, On the finite volume element method. Numer. Math.58 (1991) 713-735. · Zbl 0731.65093 · eudml:133527
[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. Zbl0729.65086 · Zbl 0729.65086 · doi:10.1137/0728022
[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. · Zbl 0707.65073 · doi:10.1137/0727039
[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). · Zbl 0990.65122
[11] Y. Coudière, J.P. Vila and P. Villedieu, Convergence rate of a finite volume scheme for a two dimensionnal diffusion convection problem. ESAIM: M2AN33 (1999) 493-516. · Zbl 0937.65116 · doi:10.1051/m2an:1999149 · publish.edpsciences.org · eudml:197561
[12] B. Courbet and J.P. Croisille, Finite volume box schemes on triangular meshes. RAIRO Modél. Math. Anal. Numér.32 (1998) 631-649. Zbl0920.65065 · Zbl 0920.65065 · eudml:193889
[13] M. Dauge, Elliptic Boundary Value Problems in Corner Domains. Lect. Notes Math., Springer-Verlag, Berlin (1988). · 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. · Zbl 0724.65093 · doi:10.2307/2008390
[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). · Zbl 0982.65122
[16] P.A. Forsyth and P.H. Sammon, Quadratic convergence for cell-centered grids. Appl. Numer. Math.4 (1988) 377-394. · Zbl 0651.65086 · doi:10.1016/0168-9274(88)90016-5
[17] B. Heinrich, Finite Difference Methods on Irregular Networks. Internat. Ser. Numer. Anal.82, Birkhaüser, Verlag Basel (1987). · 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 Equations11 (1994) 165-173. · Zbl 0822.65085 · doi:10.1002/num.1690110205
[19] F. Jacon and D. Knight, A Navier-Stokes algorithm for turbulent flows using an unstructured grid and flux difference splitting. AIAA (1994). Zbl0806.76053 · Zbl 0806.76053 · doi:10.1016/0045-7930(94)90023-X
[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. · Zbl 0913.65097 · doi:10.1137/S0036142994264699
[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. Zbl0635.65092 · Zbl 0635.65092 · doi:10.2307/2008170
[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 Equations14 (1998) 193-212. Zbl0903.65083 · Zbl 0903.65083 · doi:10.1002/(SICI)1098-2426(199803)14:2<193::AID-NUM4>3.0.CO;2-J
[25] K.W. Morton and E. Süli, Finite volume methods and their analysis. IMA J. Numer. Anal.11 (1991) 241-260. · Zbl 0729.65087 · doi:10.1093/imanum/11.2.241
[26] E. Süli, Convergence of finite volume schemes for Poisson’s equation on nonuniform meshes. SIAM J. Numer. Anal.28 (1991) 1419-1430. · Zbl 0802.65104 · doi:10.1137/0728073
[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 Equations14 (1998) 213-231. · Zbl 0903.65084 · doi:10.1002/(SICI)1098-2426(199803)14:2<213::AID-NUM5>3.0.CO;2-R
[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. · Zbl 0813.65115 · doi:10.1137/0913073
[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. Zbl0644.65062 · Zbl 0644.65062 · doi:10.1137/0725025
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.