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The accuracy of cell vertex finite volume methods on quadrilateral meshes

Author: Endre Süli
Journal: Math. Comp. 59 (1992), 359-382
MSC: Primary 65N30; Secondary 65N15, 65N50
MathSciNet review: 1134740
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Abstract: For linear first-order hyperbolic equations in two dimensions we restate the cell vertex finite volume scheme as a finite element method. On structured meshes consisting of distorted quadrilaterals, the global error is shown to be of second order in various mesh-dependent norms, provided that the quadrilaterals are close to parallelograms in the sense that the distance between the midpoints of the diagonals is of the same order as the measure of the quadrilateral. On tensor product nonuniform meshes, the cell vertex scheme coincides with the familiar box scheme. In this case, second-order accuracy is shown without any additional assumption on the regularity of the mesh, which explains the insensitivity of the cell vertex scheme to mesh stretching in the coordinate directions, observed in practice.

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  • [1] R. A. Adams, Sobolev spaces, Academic Press, New York, 1975. MR 0450957 (56:9247)
  • [2] J.-P. Aubin and I. Ekeland, Applied nonlinear analysis, Wiley-Interscience, New York, 1988. MR 964688 (89g:58001)
  • [3] P. Ciarlet, The finite element method for elliptic problems, North-Holland, Amsterdam, New York, Oxford, 1978. MR 0520174 (58:25001)
  • [4] M. Giles, Accuracy of node based solutions on irregular meshes, Proceedings of the 11th International Conference on Numerical Methods in Fluid Dynamics (D. L. Dwoyer, M. Y. Hussaini, and R. G. Voigt, eds.), Springer-Verlag, Berlin, 1988, pp. 273-277. MR 1002803 (90a:76039)
  • [5] A. Jameson, W. Schmidt, and E. Turkel, Numerical solutions of the Euler equations by finite volume methods using Runge-Kutta time-marching schemes, Paper 81-1259, AIAA, New York, 1981.
  • [6] P. Lesaint and P.-A. Raviart, Finite element collocation methods for first order systems, Math. Comp. 33 (1979), 891-918. MR 528046 (80d:65118)
  • [7] R. W. MacCormack and A. J. Paullay, Computational efficiency achieved by time splitting of finite difference operators, Paper 72-154, AIAA, New York, 1972.
  • [8] P. W. McDonald, The computation of transonic flow through two-dimensional gas turbine cascades, Paper 71-GT-89, ASME, New York, 1971.
  • [9] K. W. Morton and M. F. Paisley, A finite volume scheme with shock fitting for the steady Euler equations, J. Comp. Phys. 80 (1989), 168-203.
  • [10] K. W. Morton and E. Süli, Finite volume methods and their analysis, IMA J. Numer. Anal. 11 (1991), 241-260. MR 1105229 (93e:65145)
  • [11] R.-H. Ni, A multiple grid scheme for solving the Euler equations, AIAA J. 20 (1981), 1565-1571.
  • [12] A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, New York, 1983. MR 710486 (85g:47061)
  • [13] P. Roe, The influence of mesh quality on solution accuracy, preprint, Cranfield Institute of Technology, 1989. MR 1149110 (92k:65126)
  • [14] -, Error estimates for cell-vertex solutions of the compressible Euler equations, ICASE Report, No. 87-6, 1987.
  • [15] E. Süli, The accuracy of finite volume methods on distorted partitions, Mathematics of Finite Elements and Applications VII (J. R. Whiteman, ed.), Academic Press, 1991, pp. 253-260. MR 1132503 (92i:65171)
  • [16] -, Convergence of finite volume schemes for Poisson's equation on nonuniform meshes, SIAM J. Numer. Anal. 28 (1991), 1419-1430. MR 1119276 (92h:65159)
  • [17] M. Zlámal, Superconvergence and reduced integration in the finite element method, Math. Comp. 32 (1978), 663-685. MR 0495027 (58:13794)

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Keywords: Finite volume methods, stability, error estimates
Article copyright: © Copyright 1992 American Mathematical Society

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