Abstract
The finite volume element method (FVE) is a discretization technique for partial differential equations. This paper develops discretization energy error estimates for general self-adjoint 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 in [R. E. Bank and D. J. Rose, SIAM J. Numer. Anal., 24 (1987), pp. 777--787], [Z. Q. Cai, Numer. Math., 58 (1991), pp. 713--735] are removed. The authors finally provide a counterexample to show that an expected L2-error estimate does not exist in the usual sense. It is conjectured that the optimal order of $\|u-u_h\|_{0,\Omega}$ should be O(h) for the general case.
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