Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-05T07:59:55.301Z Has data issue: false hasContentIssue false
  • English
  • Français

Finite Volume Methods for Elliptic PDE's: A NewApproach

Published online by Cambridge University Press:  15 May 2002

Panagiotis Chatzipantelidis
Panagiotis Chatzipantelidis*
Affiliation:
Department of Applied Mathematics, University of Crete and Institute of Applied and Computational Mathematics, FO.R.T.H, 71110 Heraklion, Crete, Greece. e-mail:
Get access

Abstract

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

Keywords

Finite volume methodserror estimates
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 36 , Issue 2 , March 2002 , pp. 307 - 324
Copyright
© EDP Sciences, SMAI, 2002

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

R.A. Adams, Sobolev Spaces. Academic Press, New York (1975).
Bank, R.E. and Rose, D.J., Some error estimates for the box method. SIAM J. Numer. Anal. 24 (1987) 777-787. CrossRef
S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. Springer-Verlag, New York (1994).
Brighi, B., Chipot, M. and Gut, E., Finite differences on triangular grids. Numer. Methods Partial Differential Equations 14 (1998) 567-579. 3.0.CO;2-G>CrossRef
Z. Cai, On the finite volume element method. Numer. Math. 58 (1991) 713-735.
Champier, S., Gallouët, T. and Herbin, R., Convergence of an upstream finite volume scheme for a nonlinear hyperbolic equation on a triangular mesh. Numer. Math. 66 (1993) 139-157. CrossRef
Chatzipantelidis, P., A finite volume method based on the Crouzeix-Raviart element for elliptic PDE's in two dimensions. Numer. Math. 82 (1999) 409-432. CrossRef
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.
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).
P. Chatzipantelidis, Ch. Makridakis and M. Plexousakis, A-posteriori error estimates of a finite volume scheme for the Stokes equations. In preparation.
Chou, S.H., Analysis and convergence of a covolume method for the generalized Stokes problem. Math. Comp. 66 (1997) 85-104. CrossRef
Chou, S.H. and Error, Q. Li 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. CrossRef
P.G. Ciarlet, Basic Error Estimates for Elliptic Problems. Handbook of Numerical Analysis, Vol. II, North-Holland, Amsterdam (1991) 17-351.
Crouzeix, M. and Raviart, P.-A., Conforming and nonconforming finite element methods for solving the stationary Stokes equation I. RAIRO Anal. Numér. 7 (1973) 33-76.
Ewing, R.E., Lazarov, R.D. and Lin, Y., Finite Volume Element Approximations of Nonlocal Reactive Flows in Porous Media. Numer. Methods Partial Differential Equations 16 (2000) 285-311. 3.0.CO;2-3>CrossRef
R. Eymard, T. Gallouët and R. Herbin, Finite Volume Methods. Handbook of Numerical Analysis, Vol. VII, North-Holland, Amsterdam (2000).
P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman, Massachusetts (1985).
Hackbusch, W., On first and second order box schemes. Comput. 41 (1989) 277-296. CrossRef
Jianguo, H. and Shitong, X., On the finite volume element method for general self-adjoint elliptic problems. SIAM J. Numer. Anal. 35 (1998) 1762-1774. CrossRef
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.
Kossioris, G., Makridakis, Ch. and Souganidis, P.E., Finite volume schemes for Hamilton-Jacobi equations. Numer. Math. 83 (1999) 427-442. CrossRef
Liebau, F., The finite volume element method with quadratic basis functions. Comput. 57 (1996) 281-299. CrossRef
Mishev, I.D., Finite volume element methods for non-definite problems. Numer. Math. 83 (1999) 161-175. CrossRef
K.W. Morton, Numerical Solution of Convection-Diffusion Problems. Chapman & Hall, London (1996).
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.
H.-G. Roos, M. Stynes and L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations. Springer-Verlag, Berlin (1996).
Schmidt, T., Box schemes on quadrilateral meshes. Comput. 51 (1994) 271-292. CrossRef
R. Temam, Navier-Stokes Equations. North-Holland, Amsterdam (1979).
Weiser, A. and Wheeler, M.F., On convergence of Block-Centered finite differences for elliptic problems. SIAM J. Num. Anal. 25 (1988) 351-375. CrossRef