zbMATH — the first resource for mathematics

Convergence analysis of a finite volume method via a new nonconforming finite element method. (English) Zbl 0903.65084
The article is concerned with a convergence theory for a finite volume method and a nonconforming finite element method for Poisson’s equation (with Dirichlet boundary conditions) on a bounded convex polynomial domain in \(\mathbb{R}^2\). The finite volume method is formulated via Voronoi box partitions, while the finite element method uses the corresponding dual box partition. Since both methods amount to the same linear system of discretized equations, convergence results for the finite volume method can be obtained by convergence analysis of the finite element method. The latter is performed on the basis of the second lemma of Strang.
Reviewer: M.Plum (Karlsruhe)

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
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
Full Text: DOI
[1] Finite difference methods on irregulare networks. A generalized approach to second order problems, ISNM Vol. 82, Birkhäuser, Basel, 1987. · doi:10.1007/978-3-0348-7196-9
[2] Herbin, Numer. Meth. Part. Differ. Eq. 11 pp 165– (1995) · Zbl 0822.65085 · doi:10.1002/num.1690110205
[3] and , ”Finite volume methods for reaction-diffusion problems,” in Finite Volumes for Complex Applications, and , Eds., Hermes, Paris, 1996, pp. 231-240.
[4] Miller, IMA J. Numer. Anal. 14 pp 257– (1994) · Zbl 0806.65111 · doi:10.1093/imanum/14.2.257
[5] ”Finite volume and finite volume element methods for nonsymmetric problems,” Ph.D. thesis, Texas A&M University, 1996.
[6] Baranger, M2 AN 30 pp 445– (1996)
[7] Hackbusch, Computing 41 pp 277– (1989) · Zbl 0649.65052 · doi:10.1007/BF02241218
[8] Kerkhoven, SIAM J. Numer. Anal. 33 pp 1864– (1996) · Zbl 0860.65101 · doi:10.1137/S0036142992241824
[9] Vanselow, Computing 57 pp 93– (1996) · Zbl 0858.65109 · doi:10.1007/BF02276874
[10] ”Variational crimes in the finite element method,” in The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, Ed., Academic Press, New York, 1972, pp. 689-710. · doi:10.1016/B978-0-12-068650-6.50030-7
[11] Finite Elements, Springer, Berlin, 1992. · doi:10.1007/978-3-662-07234-9
[12] ”Basic error estimates for elliptic problems,” in Handbook of Numerical Analysis–Vol. II–Finite Element Methods (Part 1), and , Eds., Elsevier, Amsterdam, 1991, pp. 17-351. · Zbl 0875.65086 · doi:10.1016/S1570-8659(05)80039-0
[13] and , Numerik Partieller Differentialgleichungen, Teubner, Stuttgart, 1992.
[14] and , Computational Geometry–An Introduction, Springer, New York, 1985. · doi:10.1007/978-1-4612-1098-6
[15] and , ”Ein neuer Zugang zur Konvergenzanalysis einer Finite-Volumen-Methode bei der Poisson-Aufgabe,” Preprint MATH-NM-12-1996, TU Dresden (1996).
[16] Angermann, IMA J. Numer. Anal. 15 pp 161– (1995) · Zbl 0831.65117 · doi:10.1093/imanum/15.2.161
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.