*(English)*Zbl 1036.65084

The accuracy of the finite volume element (FVE) methods for solving second-order elliptic boundary value problems is studied. The approach presented herein combines traditional finite element and finite difference methods as a variation of the Galerkin finite element method, revealing regularities in the exact solution and establishing that the source term can affect the accuracy of FVE methods.

Optimal order ${H}^{1}$ and ${L}^{2}$ error estimates and superconvergence are also discussed. Some examples are given to show that FVE method cannot have the standard $O\left({h}^{2}\right)$ convergence rate in the ${L}^{2}$ norm when the source term has the minimum regularity in ${L}^{2}$, even if the exact solution is in ${H}^{2}$.

The interested reader could also refer to *R. E. Ewing*, *Z. Li*, *T. Lin* and *Y. Lin* [Math. Comput. Simul. 50 , 63–76 (1999; Zbl 1027.65155)] and *T. Kerkhoven* [SIAM J. Numer. Anal. 33, 1864–1884 (1996; Zbl 0860.65101)].

##### MSC:

65N15 | Error bounds (BVP of PDE) |

65N30 | Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE) |

35J25 | Second order elliptic equations, boundary value problems |