Summary
The finite volume element method (FVE) is a discretization technique for partial differential equations. It uses a volume integral formulation of the problem with a finite partitioning set of volumes to discretize the equations, then restricts the admissible functions to a finite element space to discretize the solution. this paper develops discretization error estimates for general selfadjoint elliptic boundary value problems with FVE based on triangulations with linear finite element spaces and a general type of control volume. We establishO(h) estimates of the error in a discreteH 1 semi-norm. Under an additional assumption of local uniformity of the triangulation the estimate is improved toO(h 2). Results on the effects of numerical integration are also included.
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References
Adams, R.A.: Sobolev spaces. Ed. New York: Academic Press 1975
Baliga, B.R., Patankar, S.V.: A new finite-element formulation for convection-diffusion problems. Numer. Heat Transfer3, 393–409 (1980)
Bank, R.E., Rose, D.J.: Some error estimates for the box method. SIAM J. Numer. Anal.24, 777–787 (1987)
Bramble, J.H., Hilbert, S.R.: Estimation of linear functionals on Sobolev spaces with application to Fourier transforms and spline interpolation. SIAM J. Numer. Anal.7, 112–124 (1970)
Cai, Z.: A theoretical foundation of the finite volume element method. Ph. D. Thesis, University of Colorado at Denver, May 1990
Cai, Z., Mandel, J., McCormick, S.: The finite volume element method for diffusion equations on general triangulations. SIAM J. Numer. Anal. (to appear)
Cai, Z., McCormick, S.: On the accuracy of the finite volume element method for diffusion equations on composite grids. SIAM J. Numer. Anal.27, 636–655 (1990)
Ciarlet, P.G.: The finite element method for elliptic problems. Amsterdam. North-Holland 1978
Ewing, R.E., Lazarov, R.D., Vassilevski, P.S.: Local refinement techniques for elliptic problems on cell-centered grids. Univ. Wyoming E.O.R.I. rep. no 1888-16
Hachbusch, W.: On first and second order box schemes. Computing41, 277–296 (1989)
Heinrich, B.: Finite difference methods on irregular networks. Basel: Birkhäuser 1987
Kadlec, J.: On the regularity of the solution of the Poisson equation on a domain with boundary locally similar to the boundary of a convex domain. Czechoslovak Math. J.14, 386–393 (1964)
Kreiss, H.O., Manteuffel, T.A., Swartz, B., Wendroff, B., White, A.B.: Supraconvergent schemes on irregular grids. Math. Comput.47, 537–554 (1986)
Manteuffel, T.A., White, A.B.: The numerical solution of second-order boundary value problems on nonuniform meshes. Math. Comput.47, 511–535 (1986)
McCormick, S., Thomas, J.: The fast adaptive composite grid method (FAC) for elliptic boundary value problems. Math. Comput.6, 439–456 (1986)
Nečas, J.: Les méthodes directes en théorie des equations elliptiques. Paris: Masson 1967
Oganesjan, A., Ruchovec, L.A.: Variational methods of solving elliptic equations (in Russian). Erevan, Izd. AN Arm. SSR, 1979
Zlámal, M.: On the finite element method. Numer. Math.12, 394–409 (1968)
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This research was sponsored in part by the Air Force Office of Scientific Research under grant number AFOSR-86-0126 and the National Science Foundation under grant number DMS-8704169. This work was performed while the author was at the University of Colorado at Denver
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Cai, Z. On the finite volume element method. Numer. Math. 58, 713–735 (1990). https://doi.org/10.1007/BF01385651
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DOI: https://doi.org/10.1007/BF01385651