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Finite volume box schemes on triangular meshes. (English) Zbl 0920.65065
The authors propose a finite element box scheme for equations of the form \(\text{div }\phi(u)= f\) generalizing that of H. B. Keller [Numerical solution partial differential equations. II: Proc. 2nd Sympos. numerical solution partial diff. equations, SYNSPADE 1970, Univ. Maryland, 327-350 (1971; Zbl 0243.65060)]. They prove an error estimate in the discrete energy seminorm for the Poisson problem. Some numerical results and implementation details are given demonstrating that the method is effectively second order.

MSC:
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
65N15 Error bounds for boundary value problems involving PDEs
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[1] R. E. BANK, D. J. ROSE, Some error estimates for the box method, SIAM J. Numer. Anal, 24, 4, 1987, 777-787. Zbl0634.65105 MR899703 · Zbl 0634.65105 · doi:10.1137/0724050
[2] [2] J. BARANGER, J. F. MAĨTRE, F. OUDIN, Connection between finite volume and mixed finite element methods, Math. Model. and Numer. Anal. (M2AN), to appear. Zbl0857.65116 MR1399499 · Zbl 0857.65116 · eudml:193811
[3] D. BRAESS, Finite Elemente, Springer Lehrbuch, 1991. Zbl0754.65084 · Zbl 0754.65084
[4] S. C. BRENNER, L. R. SCOTT, The mathematical theory of finite element methods, Texts in Applied Mathematics 15, Springer. Zbl0804.65101 · Zbl 0804.65101
[5] F. BREZZI, M. FORTIN, Mixed and Hybrid Finite Element Methods, Springer Series in Comp. Math., 15, Springer Verlag, New-York, 1991. Zbl0788.73002 MR1115205 · Zbl 0788.73002
[6] F. CASIER, H. DECONNINCK, C. HIRSCH, A class of central bidiagonal schemes with implicit boundary conditions for the solution of Euler’s equations, AIAA-83-0126, 1983.
[7] J. J. CHATTOT, S. MALET, A ”box-scheme” for the Euler equations, Lecture Notes in Math., 1270, Springer-Verlag, 1986, 52-63. Zbl0626.65088 MR910106 · Zbl 0626.65088
[8] B. COURBET, Schémas boîte en réseau triangulaire, ONERA, 1992, unpublished.
[9] B. COURBET, Schémas à deux points pour la simulation numérique des écoulements, La Recherche Aérospatiale, n^\circ 4, 1990, 21-46. Zbl0708.76105 · Zbl 0708.76105
[10] B. COURBET, Étude d’une famille de schémas boîtes à deux points et application à la dynamique des gaz monodimensionnelle, La Recherche Aérospatiale, n^\circ 5, 1991, 31-44.
[11] [11] M. CROUZEIX, P. A. RAVIART, Conforming and non conforming finite element methods for solving the stationary Stokes equations I, R.A.I.R.O. 7, 1973, R-3, 33-76. Zbl0302.65087 MR343661 · Zbl 0302.65087 · eudml:193250
[12] P. EMONOT, Méthodes de volumes-éléments-finis: Application aux équations de Navier-Stokes et résultats de convergence, Thèse de l’Université de Lyon 1, France 1992.
[13] G. FAIRWEATHER, R. D. SAYLOR, The reformulation and numerical solution of certain nonclassical initial-boundary value problems, SIAM J. Sci. Stat. Comput., 12, 1, 1991, 127-144. Zbl0722.65062 MR1078800 · Zbl 0722.65062 · doi:10.1137/0912007
[14] M. FARHLOUL, M. FORTIN, A new mixed finite element for the Stokes and elasticity problems, SIAM J. Numer. Anal., 30, 4, 1993, 971-990. Zbl0777.76051 MR1231323 · Zbl 0777.76051 · doi:10.1137/0730051
[15] W. HACKBUSCH, On first and second order box schemes, Computing, 41, 1989, 277-296. Zbl0649.65052 MR993825 · Zbl 0649.65052 · doi:10.1007/BF02241218
[16] C. JOHNSON, Adaptive finite element method for diffusion and convection problems, Comp. Meth. in Appl. Mech. Eng., 82, 1990, 301-322. Zbl0717.76078 MR1077659 · Zbl 0717.76078 · doi:10.1016/0045-7825(90)90169-M
[17] H. B. KELLER, A new difference scheme for parabolic problems, Numerical solutions of partial differential equations, II, B. Hubbard éd., Academic Press, New-York, 1971, 327-350. Zbl0243.65060 MR277129 · Zbl 0243.65060
[18] P. C. MEEK, J. NORBURY, Nonlinear moving boundary problems and a Keller box scheme, SIAM J. Numer. Anal., 21, 5, 1984, 883-893. Zbl0558.65087 MR760623 · Zbl 0558.65087 · doi:10.1137/0721057
[19] R. A. NICOLAIDES, The covolume approach to Computing incompressible flows, Incompressible Comp. Fluid Dynamics, M. P. Gunzberger, R. A. Nicolaides Ed., 1993, Cambridge Univ. Press. · Zbl 1189.76392
[20] R. A. NICOLAIDES, Direct discretization of planar div-curl problems, SIAM J. Numer. Anal., 29, 1, 1992, 32-56. Zbl0745.65063 MR1149083 · Zbl 0745.65063 · doi:10.1137/0729003
[21] R. A. NICOLAIDES, X. WU, Covolume solutions of three dimensional div-curl equations, ICASE Report 95-4. Zbl0889.35006 · Zbl 0889.35006 · doi:10.1137/S0036142994277286
[22] B. J. NOYE, Some three-level finite difference methods for simulating advection in fluids, Computers and Fluids, 19, 1991, 119-140. Zbl0721.76053 MR1087166 · Zbl 0721.76053 · doi:10.1016/0045-7930(91)90010-F
[23] P. A. RAVIART, J. M. THOMAS, A mixed finite element method for 2nd order elliptic problems, Lecture Notes in Math, 606, Springer-Verlag, 1977, 292-315. Zbl0362.65089 MR483555 · Zbl 0362.65089
[24] S. F. WORNOM, Application of compact difference schemes to the conservative Euler equations for one-dimensional flow, NASA TM 8326. · Zbl 0563.76023
[25] S. F. WORNOM, A two-point difference scheme for Computing steady-state solutions to the conservative one-dimensional Euler equations, Computers and Fluids, 12, 1, 1984, 11-30. Zbl0563.76023 · Zbl 0563.76023 · doi:10.1016/0045-7930(84)90024-0
[26] S. F. WORNOM, M. M. HAFEZ, Implicit conservative schemes for the Euler equations, AIAA J., 24, 2, 1986, 215-233. Zbl0591.76108 MR825091 · Zbl 0591.76108 · doi:10.2514/3.9248
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