zbMATH — the first resource for mathematics

Connection between finite volume and mixed finite element methods. (English) Zbl 0857.65116
Authors’ abstract: For the model problem with Laplacian operator, we show how to produce cell-centered finite volume schemes, starting from the mixed dual formulation discretized with the Raviart-Thomas element of lowest order.
The method is based on the use of an appropriate integration formula (mass lumping) allowing an explicit elimination of the vector variables. The analysis of the finite volume scheme (well-posedness and error bounds) is directly deduced from classical results of mixed finite element theory, which is the main interest of the method.
We emphasize existence and properties of the diagonalizing integration formulas, specially in the case of \(N\)-dimensional simplicial elements.
Reviewer: P.Burda (Praha)

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
Full Text: DOI EuDML
[1] J. BARANGER, P. EMONOT, J.-F. MAITRE, 1992, High order lagrangian finite volume elements for elliptic boundary value problems, Numerical Methods in Engineering 92, Ch. Hirsch et al. ed., Elsevier Science Publishers, pp. 709-713.
[2] J. BARANGER, J.-F MAITRE, F. OUDIN, 1994, Application de la théorie des éléments finis mixtes à l’étude d’une classe de schémas aux volumes différences finis pour les problèmes elliptiques, C.R. Acad. Sci. Paris, 319, Série I, pp.401-404. Zbl0804.65102 MR1289320 · Zbl 0804.65102
[3] F. BREZZI, M. FORTIN, 1991, Mixed and Hybrid Finite Element Methods, Springer Series in Computational Mathematics, 15, Springer-Verlag, New-York. Zbl0788.73002 MR1115205 · Zbl 0788.73002
[4] P. G. CIARLET, P.A. RAVIART, 1972, General Lagrange and Hermite interpolation in Rn with applications to finite element methods, Arch. Rat. Mech. Anal., 46, pp. 177-199. Zbl0243.41004 MR336957 · Zbl 0243.41004 · doi:10.1007/BF00252458
[5] I. FAILLE, 1992, A control volume method to solve an elliptic equation on a two-dimensional irregular mesh, Comput. Methods Appl. Mech. Engrg,, 100, pp. 275-290. Zbl0761.76068 MR1187634 · Zbl 0761.76068 · doi:10.1016/0045-7825(92)90186-N
[6] I. FAILLE, T. GALLOUET, R. HERBIN, 1991, Des mathématiciens découvrent les volumes finis, S.M.A.I., Matapli, Bulletin de liaison, 28.
[7] M. FARHLOUL, 1991, Méthodes d’éléments finis mixtes et volumes finis, Thèse, Université Laval, Québec.
[8] D. A. Jr. FORSYTH, P. H. SAMMON, 1988, Quadratic convergence for cell-centered grids, Applied Numerical Mathematics, 4, pp. 377-394. Zbl0651.65086 MR948505 · Zbl 0651.65086 · doi:10.1016/0168-9274(88)90016-5
[9] W. HACKBUSCH, 1989, On first and second order box schemes, Computing, 41, pp. 277-296. Zbl0649.65052 MR993825 · Zbl 0649.65052 · doi:10.1007/BF02241218
[10] Y. HAUGAZEAU, P. LACOSTE, 1993, Condensation de la matrice de masse pour les éléments finis mixtes de H (rot), C.R. Acad. Sci. Paris, 316, Série I, pp. 509-512. Zbl0767.65076 MR1209276 · Zbl 0767.65076
[11] R. HERBIN, An error estimate for a finite volume scheme for a diffusion convection problem on a triangular mesh, accepted for publication in Num. Meth. P.D.E. Zbl0822.65085 MR1316144 · Zbl 0822.65085 · doi:10.1002/num.1690110205
[12] K. W. MORTON, E. SULI, 1991, Finite volume methods and their analysis, IMA Journal of Num. Anal., 11,pp. 241-260. Zbl0729.65087 MR1105229 · Zbl 0729.65087 · doi:10.1093/imanum/11.2.241
[13] J. C. NEDELEC, 1991, Notions sur les techniques d’éléments finis, Mathématiques et Applications, 7, Ellipses-Edition Marketing. Zbl0847.65078 · Zbl 0847.65078
[14] P. A. RAVIART, J. M. THOMAS, 1977, A mixed finite element method for second order elliptic problems, Mathematical Aspects of the Finite Element Methods (I. Galligani, E. Magenes, eds), Lectures Notes in Math., 606, Springer-Verlag, New-York. Zbl0362.65089 MR483555 · Zbl 0362.65089
[15] J. E. ROBERTS, J. M. THOMAS, 1989, Mixed and hybrid methods, in Handbook of Numerical Analysis, (P. G. Ciarlet and J. L. Lions, eds.), Vol. II, Finite Element Methods (Part 1), North-Holland, Amsterdam. Zbl0875.65090 MR1115239 · Zbl 0875.65090
[16] A. WEISER, M. F. WEELER1988, On convergence of block-centered finite differences for elliptic problems, SIAM J. Numer. Anal., 25, pp.351-375. Zbl0644.65062 MR933730 · 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.