Abstract
Box schemes (finite volume methods) are widely used in fluiddynamics, especially for the solution of conservation laws. In this paper two box-schemes for elliptic equations are analysed with respect to quadrilateral meshes. Using a variational formulation, we gain stability theorems for two different box methods, namely the so-called diagonal boxes and the centre boxes. The analysis is based on an elementwise eigenvalue problem. Stability can only be guaranteed under additional assumptions on the geometry of the quadrilaterals. For the diagonal boxes unsuitable elements can lead to global instabilities. The centre boxes are more robust and differ not so much from the finite element approach. In the stable case, convergence results up to second order are proved with well-known techniques.
Zusammenfassung
Box-Methoden (Finite-Volumen-Methoden) sind verbreitete Verfahren zur Lösung physikalischer Erhaltungsgleichungen, insbesondere in der Strömungsmechanik. In dieser Arbeit werden zwei Methoden für elliptische Differentialgleichungen untersucht, die Diagonal-Boxen und die Schwerpunkt-Boxen. Da die Box-Methoden im Sinne von Petrov-Galerkin-Verfahren interpretiert werden können, erhält man vergleichbar zur Finiten-Element-Methode eine variationsrechnerische Stabilitäts- und Fehleranalyse. Damit werdenO(h)- undO(h 2)-Fehlerabschätzungen hergeleitet. Lokale Eigenwertprobleme führen zu Stabilitätsaussagen. Allerdings ergibt sich eine Abhängigkeit von der Anzahl und Art gestörter Vierecke. Insbesondere die Diagonal-Boxen sind anfällig für lokale Störungen.
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References
Albrecht, D.: Die Kopplung finiter Methoden für Transport-Diffusions-Probleme. Dissertation, Darmstadt 1989.
Aziz, A. K.: The mathematical foundation of the finite element method with applications to partial differential equations. London, New York: Academic Press 1972.
Baliga, B. R., Patankar, S. V.: A new finite-element formulation for convection-diffusion problems. Num. Heat Transfer3, 393–409 (1980).
Bank, R. E.; Rose, D. J.: Some error estimates for the box method. SIAM J. Numer. Anal.24, 777–787 (1987).
Braess, D.: Finite Elemente: Theorie, schnelle Löser und Anwendungen in der Elastizitätstheorie. Berlin, Heidelberg, New York, Tokyo: Springer 1992.
Bramble, J. H.; Hilbert, S. R.: Estimation of linear functions on Sobolev spaces with application to Fourier transforms and spline interpolation. SIAM J. Numer. Anal.7, 112–124 (1970).
Cai, Z.: A theoretical foundation for the finite volume element method. Ph. D. Thesis, University of Colorado, Denver, 1990.
Cai, Z.: On the finite volume element method. Numer. Math.58, 713–735 (1991).
Cai, Z.; Mandel, J.; McCornick, S.: The finite volume element method for diffusion equations on general triangulations. SIAM J. Numer. Anal.28, 392–402 (1991).
Ciarlet, P. G.: The finite element method for elliptic problems. Amsterdam, New York, Oxford: North-Holland Publ. Comp. 1978.
Dick, E.: Introduction to finite volume techniques in computational fluid dynamics. State University of Ghent, Report, 1990.
Girault, V.: Theory of a finite difference method on irregular networks. SIAM J. Numer. Anal.11, 260–282 (1974).
Hackbusch, W.: Elliptic differential equations: theory and numerical treatment. Berlin, Heidelberg, New York, Tokyo: Springer 1992.
Hackbusch, W.: On first and second order box schemes, Computing41, 277–296 (1989).
Heinrich, B.: Finite difference methods on irregular networks, ISNM vol. 82. Basel, Boston, Stuttgart: Birkhauser Verlag 1987.
Liebau, F.: Analyse einer Finite-Volumen-Elemente-Methode mit quadratischen Ansatzfunktionen. Doctoral Thesis, Kiel, 1992.
Mackenzie, J. A.; Morton, K. W.: Finite volume solutions of convection-diffusion test problems. Math. Comp.60, 189–220 (1993).
Morton, K. W.; Süli, E.: Finite volume methods and their analysis. IMA J. Numer. Anal.,11 241–260 (1991).
Patankar, S.: Numerical heat transfer and fluid flow. New York: Hemisphere 1980.
Schmidt, T.: Analyse zweier Finite-Volumen Methoden für elliptische partielle Differentialgleichungen 2. Ordnung auf Vierecksgittern. Doctoral Thesis, Kiel, 1992.
Süli, E.: The accuracy of cell vertex finite volume methods on quadrilateral meshes. Math. Comp.59, 359–382 (1992).
Süli, E.: Convergence of finite volume schemes for Poisson's equation on non-uniform meshes. SIAM J. Numer. Anal.28 (1991).
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Schmidt, T. Box schemes on quadrilateral meshes. Computing 51, 271–292 (1993). https://doi.org/10.1007/BF02238536
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DOI: https://doi.org/10.1007/BF02238536