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 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 L-2-error estimate does not exist in the usual sense. It is conjectured that the optimal order of parallel to u - u(h) parallel to(0,Omega) should be O(h) for the general case. [References: 9]
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机译：有限体积元方法（FVE）是偏微分方程的离散化技术。本文基于三角剖分，开发了具有FVE的一般自伴椭圆椭圆边值问题的离散化能量误差估计，该三角函数上存在线性有限元空间，以及一种非常通用的控制体积（体积）。本文的能量误差估计也是最佳的，但在[R. E. Bank和D. J. Rose，SIAM J. Numer。 Anal。，24（1987），pp.777-787]，[Z.蔡，Numer。 Math。，58（1991），pp.713-735]。作者最后提供了一个反例，以显示通常意义上不存在预期的L-2-误差估计。据推测，在一般情况下，平行于u-u（h）平行于（0，Omega）的最佳阶数应为O（h）。 [参考：9]
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