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On the finite volume element method for general self-adjoint elliptic problems. (English) Zbl 0913.65097

The finite volume element method (FVE) is a discretization technique for partial differential equations. This paper develops discretization energy error estimates for general selfadjoint elliptic boundary value problems with FVE based on triangulations, on which there exist linear finite element spaces, and a very general type of control volumes (covolumes).

The energy error estimates of this paper are also optimal but the restriction conditions for the covolumes given by R. E. Bank and D. J. Rose [SIAM J. Numer. Anal. 24, 777-787 (1987; Zbl 0634.65105)] and by Z. Cai [Numer. Math. 58, No. 7, 713-735 (1991; Zbl 0731.65093)] are removed. The authors finally provide a counterexample to show that an expected ${L}^{2}$-error estimate does not exist in the usual sense. It is conjectured that the optimal order of $\parallel u-{u}_{h}{\parallel }_{0,{\Omega }}$ should be $O\left(h\right)$ for the general case.

MSC:
 65N30 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE) 65N15 Error bounds (BVP of PDE) 35J25 Second order elliptic equations, boundary value problems