Finite volume methods for elliptic PDE’s : a new approach

Panagiotis Chatzipantelidis

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2002)

  • Volume: 36, Issue: 2, page 307-324
  • ISSN: 0764-583X

Abstract

top
We consider a new formulation for finite volume element methods, which is satisfied by known finite volume methods and it can be used to introduce new ones. This framework results by approximating the test function in the formulation of finite element method. We analyze piecewise linear conforming or nonconforming approximations on nonuniform triangulations and prove optimal order H 1 - norm and L 2 - norm error estimates.

How to cite

top

Chatzipantelidis, Panagiotis. "Finite volume methods for elliptic PDE’s : a new approach." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 36.2 (2002): 307-324. <http://eudml.org/doc/244691>.

@article{Chatzipantelidis2002,
abstract = {We consider a new formulation for finite volume element methods, which is satisfied by known finite volume methods and it can be used to introduce new ones. This framework results by approximating the test function in the formulation of finite element method. We analyze piecewise linear conforming or nonconforming approximations on nonuniform triangulations and prove optimal order $H^1-$norm and $L^2-$norm error estimates.},
author = {Chatzipantelidis, Panagiotis},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {finite volume methods; error estimates; second order elliptic equation; Petrov-Galerkin method; finite elements},
language = {eng},
number = {2},
pages = {307-324},
publisher = {EDP-Sciences},
title = {Finite volume methods for elliptic PDE’s : a new approach},
url = {http://eudml.org/doc/244691},
volume = {36},
year = {2002},
}

TY - JOUR
AU - Chatzipantelidis, Panagiotis
TI - Finite volume methods for elliptic PDE’s : a new approach
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2002
PB - EDP-Sciences
VL - 36
IS - 2
SP - 307
EP - 324
AB - We consider a new formulation for finite volume element methods, which is satisfied by known finite volume methods and it can be used to introduce new ones. This framework results by approximating the test function in the formulation of finite element method. We analyze piecewise linear conforming or nonconforming approximations on nonuniform triangulations and prove optimal order $H^1-$norm and $L^2-$norm error estimates.
LA - eng
KW - finite volume methods; error estimates; second order elliptic equation; Petrov-Galerkin method; finite elements
UR - http://eudml.org/doc/244691
ER -

References

top
  1. [1] R.A. Adams, Sobolev Spaces. Academic Press, New York (1975). Zbl0314.46030MR450957
  2. [2] R.E. Bank and D.J. Rose, Some error estimates for the box method. SIAM J. Numer. Anal. 24 (1987) 777–787. Zbl0634.65105
  3. [3] S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. Springer–Verlag, New York (1994). Zbl0804.65101
  4. [4] B. Brighi, M. Chipot and E. Gut, Finite differences on triangular grids. Numer. Methods Partial Differential Equations 14 (1998) 567–579. Zbl0926.65104
  5. [5] Z. Cai, On the finite volume element method. Numer. Math. 58 (1991) 713–735. Zbl0731.65093
  6. [6] S. Champier, T. Gallouët and R. Herbin, Convergence of an upstream finite volume scheme for a nonlinear hyperbolic equation on a triangular mesh. Numer. Math. 66 (1993) 139–157. Zbl0801.65089
  7. [7] P. Chatzipantelidis, A finite volume method based on the Crouzeix–Raviart element for elliptic PDE’s in two dimensions. Numer. Math. 82 (1999) 409–432. Zbl0942.65131
  8. [8] P. Chatzipantelidis, R.D. Lazarov and V. Thomée, Error estimates for the finite volume element method for parabolic pde’s in convex polygonal domains. In preparation. Zbl1067.65092
  9. [9] P. Chatzipantelidis and R.D. Lazarov, The finite volume element method in nonconvex polygonal domains. To appear in Proceedings of the Third International Symposium on Finite Volumes for Complex Applications, Hermes Science Publications, Paris (2002). Zbl1118.65385MR2007413
  10. [10] P. Chatzipantelidis, Ch. Makridakis and M. Plexousakis, A-posteriori error estimates of a finite volume scheme for the Stokes equations. In preparation. 
  11. [11] S.H. Chou, Analysis and convergence of a covolume method for the generalized Stokes problem. Math. Comp. 66 (1997) 85–104. Zbl0854.65091
  12. [12] S.H. Chou and Q. Li, Error estimates in L 2 , H 1 and L in covolume methods for elliptic and parabolic problems: a unified approach. Math. Comp. 69 (2000) 103–120. Zbl0936.65127
  13. [13] P.G. Ciarlet, Basic Error Estimates for Elliptic Problems. Handbook of Numerical Analysis, Vol. II, North–Holland, Amsterdam (1991) 17–351. Zbl0875.65086
  14. [14] M. Crouzeix and P.–A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equation I. RAIRO Anal. Numér. 7 (1973) 33–76. Zbl0302.65087
  15. [15] R.E. Ewing, R.D. Lazarov and Y. Lin, Finite Volume Element Approximations of Nonlocal Reactive Flows in Porous Media. Numer. Methods Partial Differential Equations 16 (2000) 285–311. Zbl0961.76050
  16. [16] R. Eymard, T. Gallouët and R. Herbin, Finite Volume Methods. Handbook of Numerical Analysis, Vol. VII, North–Holland, Amsterdam (2000). Zbl0981.65095
  17. [17] P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman, Massachusetts (1985). Zbl0695.35060MR775683
  18. [18] W. Hackbusch, On first and second order box schemes. Comput. 41 (1989) 277–296. Zbl0649.65052
  19. [19] H. Jianguo and X. Shitong, On the finite volume element method for general self–adjoint elliptic problems. SIAM J. Numer. Anal. 35 (1998) 1762–1774. Zbl0913.65097
  20. [20] S. Kang and D.Y. Kwak, Error estimate in L 2 of a covolume method for the generalized Stokes Problem. Proceedings of the eight KAIST Math Workshop on Finite Element Method, KAIST (1997) 121–139. 
  21. [21] G. Kossioris, Ch. Makridakis and P.E. Souganidis, Finite volume schemes for Hamilton–Jacobi equations. Numer. Math. 83 (1999) 427–442. Zbl0938.65089
  22. [22] F. Liebau, The finite volume element method with quadratic basis functions. Comput. 57 (1996) 281–299. Zbl0866.65074
  23. [23] I.D. Mishev, Finite volume element methods for non-definite problems. Numer. Math. 83 (1999) 161–175. Zbl0938.65131
  24. [24] K.W. Morton, Numerical Solution of Convection–Diffusion Problems. Chapman & Hall, London (1996). Zbl0861.65070
  25. [25] M. Plexousakis and G.E. Zouraris, High–order locally conservative finite volume-type approximations of one dimensional elliptic problems. Technical Report, TRITA-NA-0138, NADA, Royal Institute of Technology, Sweden. 
  26. [26] H.-G. Roos, M. Stynes and L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations. Springer–Verlag, Berlin (1996). Zbl0844.65075
  27. [27] T. Schmidt, Box schemes on quadrilateral meshes. Comput. 51 (1994) 271–292. Zbl0787.65075
  28. [28] R. Temam, Navier–Stokes Equations. North–Holland, Amsterdam (1979). Zbl0426.35003
  29. [29] A. Weiser and M.F. Wheeler, On convergence of Block-Centered finite differences for elliptic problems. SIAM J. Num. Anal. 25 (1988) 351–375. Zbl0644.65062

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.