Finite volume methods for elliptic PDE's : a new approach
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 36 (2002) no. 2, pp. 307-324.

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.

DOI : https://doi.org/10.1051/m2an:2002014
Classification : 65N30,  65N15
Mots clés : finite volume methods, error estimates
@article{M2AN_2002__36_2_307_0,
author = {Chatzipantelidis, Panagiotis},
title = {Finite volume methods for elliptic PDE's : a new approach},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
pages = {307--324},
publisher = {EDP-Sciences},
volume = {36},
number = {2},
year = {2002},
doi = {10.1051/m2an:2002014},
zbl = {1041.65087},
language = {en},
url = {www.numdam.org/item/M2AN_2002__36_2_307_0/}
}
Chatzipantelidis, Panagiotis. Finite volume methods for elliptic PDE's : a new approach. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 36 (2002) no. 2, pp. 307-324. doi : 10.1051/m2an:2002014. http://www.numdam.org/item/M2AN_2002__36_2_307_0/

[1] R.A. Adams, Sobolev Spaces. Academic Press, New York (1975). | MR 450957 | Zbl 0314.46030

[2] R.E. Bank and D.J. Rose, Some error estimates for the box method. SIAM J. Numer. Anal. 24 (1987) 777-787. | Zbl 0634.65105

[3] S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. Springer-Verlag, New York (1994). | Zbl 0804.65101

[4] B. Brighi, M. Chipot and E. Gut, Finite differences on triangular grids. Numer. Methods Partial Differential Equations 14 (1998) 567-579. | Zbl 0926.65104

[5] Z. Cai, On the finite volume element method. Numer. Math. 58 (1991) 713-735. | Zbl 0731.65093

[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. | Zbl 0801.65089

[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. | Zbl 0942.65131

[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. | Zbl 1067.65092

[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). | MR 2007413 | Zbl 1118.65385

[10] P. Chatzipantelidis, Ch. Makridakis and M. Plexousakis, A-posteriori error estimates of a finite volume scheme for the Stokes equations. In preparation.

[11] S.H. Chou, Analysis and convergence of a covolume method for the generalized Stokes problem. Math. Comp. 66 (1997) 85-104. | Zbl 0854.65091

[12] S.H. Chou and Q. Li, Error estimates in ${L}^{2}$, ${H}^{1}$ and ${L}^{\infty }$ in covolume methods for elliptic and parabolic problems: a unified approach. Math. Comp. 69 (2000) 103-120. | Zbl 0936.65127

[13] P.G. Ciarlet, Basic Error Estimates for Elliptic Problems. Handbook of Numerical Analysis, Vol. II, North-Holland, Amsterdam (1991) 17-351. | Zbl 0875.65086

[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. | Numdam | Zbl 0302.65087

[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. | Zbl 0961.76050

[16] R. Eymard, T. Gallouët and R. Herbin, Finite Volume Methods. Handbook of Numerical Analysis, Vol. VII, North-Holland, Amsterdam (2000). | Zbl 0981.65095

[17] P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman, Massachusetts (1985). | MR 775683 | Zbl 0695.35060

[18] W. Hackbusch, On first and second order box schemes. Comput. 41 (1989) 277-296. | Zbl 0649.65052

[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. | Zbl 0913.65097

[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] G. Kossioris, Ch. Makridakis and P.E. Souganidis, Finite volume schemes for Hamilton-Jacobi equations. Numer. Math. 83 (1999) 427-442. | Zbl 0938.65089

[22] F. Liebau, The finite volume element method with quadratic basis functions. Comput. 57 (1996) 281-299. | Zbl 0866.65074

[23] I.D. Mishev, Finite volume element methods for non-definite problems. Numer. Math. 83 (1999) 161-175. | Zbl 0938.65131

[24] K.W. Morton, Numerical Solution of Convection-Diffusion Problems. Chapman & Hall, London (1996). | Zbl 0861.65070

[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] H.-G. Roos, M. Stynes and L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations. Springer-Verlag, Berlin (1996). | Zbl 0844.65075

[27] T. Schmidt, Box schemes on quadrilateral meshes. Comput. 51 (1994) 271-292. | Zbl 0787.65075

[28] R. Temam, Navier-Stokes Equations. North-Holland, Amsterdam (1979). | Zbl 0426.35003

[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. | Zbl 0644.65062