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

Chatzipantelidis, Panagiotis

ESAIM: 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.

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

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

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

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

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

[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

[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

[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

[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 | Zbl

[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

[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

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

[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

[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

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

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

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

[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

[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

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

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

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

[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

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

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

[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