Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-22T16:19:16.706Z Has data issue: false hasContentIssue false
  • English
  • Français

Admissible Regions for Higher-Order Finite Volume Method Grids

Part of: Partial differential equations, boundary value problems

Published online by Cambridge University Press:  02 May 2017

Yuanyuan Zhang  and
Zhongying Chen
Yuanyuan Zhang*
Affiliation:
Department of Mathematics and Information Science, Yantai University, Yantai 264005, China
Zhongying Chen*
Affiliation:
Guangdong Province Key Laboratory of Computational Science, School of Mathematics and Computational Sciences, Sun Yat-sen University, Guangzhou 510275, China
*
*Corresponding author. Email addresses:[email protected] (Y. Zhang), [email protected] (Z. Chen)
*Corresponding author. Email addresses:[email protected] (Y. Zhang), [email protected] (Z. Chen)
Get access

Abstract

Admissible regions for higher-order finite volume method (FVM) grids are considered. A new Hermite quintic FVM and a new hybrid quintic FVM are constructed to solve elliptic boundary value problems, and the corresponding admissible regions are investigated. A sufficient condition for the uniform local-ellipticity of the new hybrid quintic FVM is obtained when its admissible region is known. In addition, the admissible regions for a large number of higher-order FVMs are provided. For the same class of FVM (Lagrange, Hermite or hybrid), the higher order FVM has a smaller admissible region such that stronger geometric restrictions are required to guarantee its uniform local-ellipticity.

Keywords

Finite volume methodadmissible regionuniform local-ellipticity

MSC classification

Secondary: 65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 65N12: Stability and convergence of numerical methods
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 7 , Issue 2 , May 2017 , pp. 269 - 285
Copyright
Copyright © Global-Science Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Babuška, I. and Aziz, A.K., Lectures on themathematical foundations of the finite element method, University of Maryland, College Park, Washington DC, Technical Note BN-748 (1972).Google Scholar
[2] Bank, R.E. and Rose, D.J., Some error estimates for the box method, SIAM J. Numer. Anal. 24, 777–787 (1987).CrossRefGoogle Scholar
[3] Cai, Z., On the finite volume element method, Numer. Math. 58, 713–735 (1991).Google Scholar
[4] Cai, Z., Douglas, J. and Park, M., Development and analysis of higher order finite volume methods over rectangles for elliptic equations, Adv. Comput. Math. 19, 3–33 (2013).CrossRefGoogle Scholar
[5] Chen, L., A new class of high order finite volume methods for second order elliptic equations, SIAM J. Numer. Anal. 47, 4021–4043 (2010).Google Scholar
[6] Chen, Z., The error estimate of generalized differencemethods of 3rd-order Hermite type for elliptic partial differential equations, Northeast. Math. 8, 127–135 (1992).Google Scholar
[7] Chen, Z., Wu, J. and Xu, Y., Higher-order finite volume methods for elliptic boundary value problem, Adv. Comput. Math. 37, 191–253 (2012).CrossRefGoogle Scholar
[8] Chen, Z., Xu, Y. and Zhang, Y., A construction of higher-order finite volume methods, Math. Comp., S 0025-5718 02891-3 (2014).Google Scholar
[9] Chen, Z., Xu, Y. and Zhang, Y., Higher-order finite volume methods II: Inf-sup condition and uniform local-ellipticity, J. Comput. Appl. Math. 265, 96–109 (2014).CrossRefGoogle Scholar
[10] Chou, S.-H. and Kwak, D.-Y., Multigrid algorithms for a vertex-centered covolume method for elliptic problems, Numer. Math. 90, 441–458 (2002).Google Scholar
[11] Ciarlet, P.G., The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam (1978).Google Scholar
[12] D, Y. and Li, Y. The finite element method for elliptic problems, J. Syst. Sci. Complex. 24, 991–1006 (2011).Google Scholar
[13] Ewing, R., Lin, T. and Lin, Y., On the accuracy of the finite volume element method based on piecewise linear polynomials, SIAM J. Numer. Anal. 39, 1865–1888 (2002).Google Scholar
[14] Eymard, R., Gallouët, T. and Herbin, R., Finite Volume Methods in Handbook of Numerical Analysis Vol. VII, pp. 713–1020, Ciarlet, P.G. and Lions, J.L., Eds., North-Holland, Amsterdam (2000).Google Scholar
[15] Faille, I., A control volume method to solve an elliptic equation on a two-dimensional irregular mesh, Comput. Methods Appl. Mech. Eng. 100, 275–290 (1992).Google Scholar
[16] Hackbusch, W., On first and second order box schemes, Computing 41, 277–296 (1989).CrossRefGoogle Scholar
[17] Li, R., Generalized difference methods for two point boundary value problems (in Chinese), Acta Sci. Natur. Univ. Jilin 1, 26–40 (1982).Google Scholar
[18] Li, R., Generalized difference methods for a nonlinear Dirichlet problem, SIAM J. Numer. Anal. 24, 77–88 (1987).Google Scholar
[19] Li, R., Chen, Z. and Wu, W., Generalized Difference Methods for Differential Equations: Numerical Analysis of Finite Volume Methods, Marcel Dekker, New York (2000).CrossRefGoogle Scholar
[20] Li, Y., Shu, S., Xu, Y. and Zou, Q., Multilevel preconditioning for the finite volume method, Math. Comp. 81, 1399–1428 (2012).Google Scholar
[21] Liebau, F., The finite volume element method with quadratic basis functions, Computing 57, 281–299 (1996).Google Scholar
[22] Ollivier-Gooch, C. and Altena, M.V., A high-order-accurate unstructured mesh finite-volume scheme for the advection-diffusion equation, J. Comput. Phys. 181, 729–752 (2010).CrossRefGoogle Scholar
[23] Xu, J., and Zou, Q., Analysis of linear and quadratic simplicial finite volume methods for elliptic equations, Numer. Math. 111, 469–492 (2009).Google Scholar
[24] Yu, C., Wang, Y. and Li, Y., The finite volume element methods for Poisson equation based on Adini's element, Appl. Math. Comput. 219, 5537–5549 (2013).Google Scholar
[25] Zhang, Z. and Zou, Q., A family of finite volume schemes of arbitrary order on rectangular meshes, J. Sci. Comput. 58, 308–330 (2014).Google Scholar
[26] Zhang, Z. and Zou, Q., Vertex centered finite volume schemes of any order over quadrilateral meshes for elliptic boundary value problems, Numer. Math., 130(2), 363–393 (2015).CrossRefGoogle Scholar