×

zbMATH — the first resource for mathematics

Finite volume approximation of elliptic problems and convergence of an approximate gradient. (English) Zbl 0982.65122
The authors extend earlier work [for example in Numer. Math. 82, No. 1, 91-116 (1999; Zbl 0930.65118)] on the convergence of finite volume solution methods for second order partial differential equations. The current work adds convergence of the solution gradient to earlier proofs of convergence of the solution alone.
The particular finite volume method under study requires an “admissible” mesh that depends on the line connecting the centers of adjacent mesh cells being perpendicular to the common edge of the cells. Triangular and Voronoi meshes are examples of admissible meshes. Degrees of freedom of the solution are then defined not only for mesh cells, but also for mesh edges. The mesh edge degrees of freedom are then used to define the gradient of the solution. For a mesh cell \(K\) with one edge \(\sigma\), the solution is regarded as constant over \(K\), but its gradient is regarded as being defined in terms of functions \(\phi\) that satisfy a Neumann problem with normal derivative equal to 1 on \(\sigma\) and zero on the other edges of \(K\). The properties of \(\phi\) play a crucial role in the convergence estimates.
Two numerical examples are presented for the Dirichlet problem for Poisson’s equation. The first involves a spatially variable density and is posed on a square two-dimensional domain, and the second involves Poisson’s equation on an “\(L\)”-shaped domain.

MSC:
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Agouzal, A.; Baranger, J.; Maitre, J.-F.; Oudin, F., Connection between finite volume and mixed finite element methods for a diffusion problem with non constant coefficients, with application to convection diffusion, East – west J. numer. math., 3, 4, 237-254, (1995) · Zbl 0839.65116
[2] Arbogast, T.; Wheeler, M.F.; Yotov, I., Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite differences, SIAM J. numer. anal., 34, 2, 828-852, (1997) · Zbl 0880.65084
[3] Bank, R.E.; Rose, D.J., Error estimates for the box method, SIAM J. numer. anal., 777-790, (1986)
[4] Baranger, J.; Maitre, J.-F.; Oudin, F., Connection between finite volume and mixed finite element methods, Modél. math. anal. numér., 30, 3,4, 444-465, (1996) · Zbl 0857.65116
[5] Cai, Z., On the finite volume element method, Numer. math., 58, 713-735, (1991) · Zbl 0731.65093
[6] Cai, Z.; Mandel, J.; Mc Cormick, S., The finite volume element method for diffusion equations on general triangulations, SIAM J. numer. anal., 28, 2, 392-402, (1991) · Zbl 0729.65086
[7] Y. Coudière, T. Gallouët, R. Herbin, Discrete Sobolev inequalities and \(L\^{}\{p\}\) error estimates for approximate finite volume solutions of convection diffusion equations (1998), submitted
[8] Coudière, Y.; Vila, J.P.; Villedieu, P., Convergence of a finite volume scheme for a diffusion problem, (), 161-168
[9] Coudière, Y.; Vila, J.P.; Villedieu, P., Convergence rate of a finite volume scheme for a two dimensional convection diffusion problem, Modél. math. anal. numér., 33, 3, 493-516, (1999) · Zbl 0937.65116
[10] Dubois, F., Finite volumes and mixed Petrov Galerkin finite elements: the unidimensional problem, Numer. methods partial differential equations, 16, 3, 335-360, (2000) · Zbl 0954.65062
[11] R. Eymard, T. Gallouët, R. Herbin, The finite volume method, to appear in: P.G. Ciarlet, J.L. Lions (Eds.), Handbook of Numerical Analysis
[12] R. Eymard, T. Gallouët, M. Gutnik, R. Herbin, D. Hilhorst, Finite volume methods for the approximation of the solutions to the Richards equation (1998), submitted
[13] Eymard, R.; Gallouët, T.; Herbin, R., Convergence of finite volume approximations to the solutions of semilinear convection diffusion reaction equations, Numer. math., 82, 91-116, (1999) · Zbl 0930.65118
[14] Feistauer, M.; Felcman, J.; Lukacova-Medvidova, M., On the convergence of a combined finite volume-finite element method for nonlinear convection – diffusion problems, Numer. methods partial differential equations, 13, 163-190, (1997) · Zbl 0869.65057
[15] Forsyth, P.A.; Sammon, P.H., Quadratic convergence for cell-centered grids, Appl. numer. math., 4, 377-394, (1988) · Zbl 0651.65086
[16] T. Gallouët, R. Herbin, Finite volume approximation of elliptic problems with irregular data, in: F. Benkhaldoun, R. Vilsmeier, Hänel (Eds.), Finite Volumes for Complex Applications, Problems and Perspectives II, Hermes, Paris, pp. 155-162 · Zbl 1052.65552
[17] T. Gallouët, R. Herbin, M.H. Vignal, Error estimate for the approximate finite volume solutions of convection diffusion equations with Dirichlet, Neumann or Fourier boundary conditions, accepted for publication in SIAM J. Numer. Anal
[18] Heinrich, B., Finite difference methods on irregular networks, International series of numerical mathematics, 82, (1986), Birkhäuser
[19] Herbin, R., An error estimate for a finite volume scheme for a diffusion – convection problem on a triangular mesh, Numer. methods partial differential equations, 11, 165-173, (1995) · Zbl 0822.65085
[20] Herbin, R., Finite volume methods for diffusion convection equations on general meshes, (), 153-160
[21] Herbin, R.; Labergerie, O., Finite volume schemes for elliptic and elliptic – hyperbolic problems on triangular meshes, Comput. methods appl. mech. engrg., 147, 85-103, (1997) · Zbl 0897.76072
[22] Lazarov, R.D.; Mishev, I.D., Finite volume methods for reaction diffusion problems, (), 233-240
[23] Lazarov, R.D.; Mishev, I.D.; Vassilevski, P.S., Finite volume methods for convection – diffusion problems, SIAM J. numer. anal., 33, 31-55, (1996) · Zbl 0847.65075
[24] Mackenzie, J.A.; Morton, K.W., Finite volume solutions of convection – diffusion test problems, Math. comp., 60, 201, 189-220, (1992) · Zbl 0797.76072
[25] Manteuffel, T.; White, A.B., The numerical solution of second order boundary value problem on non uniform meshes, Math. comp., 47, 511-536, (1986) · Zbl 0635.65092
[26] Mishev, I.D., Finite volume methods on Voronoi meshes, Numer. methods partial differential equations, 14, 2, 193-212, (1998) · Zbl 0903.65083
[27] Morton, K.W., Numerical solutions of convection – diffusion problems, (1996), Chapman and Hall London · Zbl 0861.65070
[28] Morton, K.W.; Süli, E., Finite volume methods and their analysis, IMA J. numer. anal., 11, 241-260, (1991) · Zbl 0729.65087
[29] Morton, K.W.; Stynes, M.; Süli, E., Analysis of a cell-vertex finite volume method for a convection – diffusion problems, Math. comp., 66, 220, 1389-1406, (1997) · Zbl 0885.65121
[30] Patankar, S.V., Numerical heat transfer and fluid flow, Series in computational methods in mechanics and thermal sciences, (1980), McGraw Hill New York
[31] Roberts, J.E.; Thomas, J.M., Mixed and hybrids methods, (), 523-640
[32] Samarskii, A.A., On monotone difference schemes for elliptic and parabolic equations in the case of a non-selfadjoint elliptic operator, Zh. vychisl. mat. i mat. fiz., 5, 548-551, (1965), (in Russian)
[33] Samarskii, A.A., Introduction to the theory of difference schemes, (1971), Nauka Moscow, (in Russian) · Zbl 0971.65076
[34] Samarskii, A.A.; Lazarov, R.D.; Makarov, V.L., Difference schemes for differential equations having generalized solutions, (1987), Vysshaya Shkola Publishers Moscow, (in Russian)
[35] Shashkov, M., Conservative finite-difference methods on general grids, (1996), CRC Press New York · Zbl 0844.65067
[36] Süli, E., The accuracy of cell vertex finite volume methods on quadrilateral meshes, Math. comp., 59, 200, 359-382, (1992) · Zbl 0767.65072
[37] Tichonov, A.N.; Samarskii, A.A., Homogeneous difference schemes on nonuniform nets, Zh. vychisl. mat. i mat. fiz., 2, 812-832, (1962), (in Russian)
[38] Vanselow, R., Relations between FEM and FVM, (), 217-223 · Zbl 0858.65109
[39] Vanselow, R.; Scheffler, Convergence analysis of a finite volume method via a new nonconforming finite element method, Numer. methods partial differential equations, 14, 213-231, (1998) · Zbl 0903.65084
[40] Vassilevski, P.S.; Petrova, S.I.; Lazarov, R.D., Finite difference schemes on triangular cell-centered grids with local refinement, SIAM J. sci. statist. comput., 13, 6, 1287-1313, (1992) · Zbl 0813.65115
[41] Weiser, A.; Wheeler, M.F., On convergence of block-centered finite-differences for elliptic problems, SIAM J. numer. anal., 25, 351-375, (1988) · Zbl 0644.65062
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.