  This paper is devoted to analyze of nonconforming finite volume methods
(FVMs), whose trial spaces are chosen as the nonconforming finite element (FE)
spaces, for solving the second order elliptic boundary value problems. We
formulate the nonconforming FVMs as special types of Petrov-Galerkin methods
and develop a general convergence theorem, which serves as a guide for the
analysis of the nonconforming FVMs. As special examples, we shall present the
triangulation based Crouzeix-Raviart (C-R) FVM as well as the rectangle mesh
based hybrid Wilson FVM. Their optimal error estimates in the mesh dependent
$H^1$-norm will be obtained under the condition that the primary mesh is
regular. For the hybrid Wilson FVM, we prove that it enjoys the same optimal
error order in the $L^2$-norm as that of the Wilson FEM. Numerical experiments
are also presented to confirm the theoretical results.

Numerical Analysis