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A generalization of the vertex-centered finite volume scheme to arbitrary high order

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Computing and Visualization in Science

Abstract

A higher order finite volume method for elliptic problems is proposed for arbitrary order \({p \in \mathbb{N}}\) . Piecewise polynomial basis functions are used as trial functions while the control volumes are constructed by a vertex-centered technique. The discretization is tested on numerical examples utilizing triangles and quadrilaterals in 2D. In these tests the optimal error is achieved in the H 1-norm. The error in the L 2-norm is one order below optimal for even polynomial degrees and optimal for odd degrees.

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References

  1. Bank R.E., Rose D.: Some error estimates for the box method. SIAM J. Numer. Anal. 24(4), 777–787 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  2. Cai Z.: On the finite volume element method. Numerische Mathematik 58(1), 713–735 (1990)

    Article  Google Scholar 

  3. Ciarlet P.: The Finite Element Method for Elliptic Problems, vol. 58. North-Holland, Amsterdam (1978)

  4. Ewing R.E., Lin T., Lin Y.: On the accuracy of the finite volume element method based on piecewise linear polynomials. SIAM J. Numer. Anal. 39(6), 1865–1888 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  5. Eymard R., Gallouet T., Herbin R.: Finite Volume Methods, vol. 7. North Holland, Amsterdam (2000)

  6. Hackbusch W.: On first and second order box schemes. Computing 41, 277–296 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  7. Knabner, P., Angermann, L.: Numerik partieller Differentialgleichungen. Springer, Berlin; Heidelberg [u.a.] (2000)

  8. Li R., Chen Z., Wu W.: Generalized Difference Methods for Differential Equations, vol. 226. Dekker, NY (2000)

    Google Scholar 

  9. Liebau F.: The finite volume element method with quadratic basis functions. Computing 57(4), 281–299 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  10. Mishev I.: Finite volume element methods for non-definite problems. Numerische Mathematik 83(1), 161–175 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  11. Süli, E.: The accuracy of cell vertex finite volume methods on quadrilateral meshes. Math. Comput. 359–382 (1992)

  12. Vogel, A.: Ein Finite-Volumen-Verfahren höherer Ordnung mit Anwendung in der Biophysik. Master’s thesis, University of Heidelberg (2008)

  13. Xu, J., Zou, Q.: Analysis of linear and quadratic finite volume methods for elliptic equations. Preprint AM298, Math. Dept., Penn State, (2005)

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Correspondence to Andreas Vogel.

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Vogel, A., Xu, J. & Wittum, G. A generalization of the vertex-centered finite volume scheme to arbitrary high order. Comput. Visual Sci. 13, 221–228 (2010). https://doi.org/10.1007/s00791-010-0139-z

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  • DOI: https://doi.org/10.1007/s00791-010-0139-z

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