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In this article, we introduce three schemes for the Poisson problem in 2D on triangular meshes, generalizing the FVbox scheme introduced by Courbet and Croisille [1]. In this kind of scheme, the approximation is performed on the mixed form of the problem, but contrary to the standard mixed method, with a pair of trial spaces different from the pair of test spaces. The latter is made of Galerkin‐discontinuous spaces on a unique mesh. The first scheme uses as trial spaces the P1nonconforming space of Crouzeix‐Raviart both for uand for the flux p= ∇u. In the two others, the quadratic nonconforming space of Fortin and Soulie is used. An important feature of all these schemes is that they are equivalent to a first scheme in uonly and an explicit representation formula for the flux p= ∇u. The numerical analysis of the schemes is performed using this property. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 355–373, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.10003

We present the numerical analysis on the Poisson problem of two mixed Petrov-Galerkin finite volume schemes for equations in divergence form $\mathop{ div}\nolimits\varphi(u,\nabla u)=f$ . The first scheme, which has been introduced in [CITE], is a generalization in two dimensions of Keller's box-scheme. The second scheme is the dual of the first one, and is a cell-centered scheme for $u$ and the flux $\varphi$ . For the first scheme, the two trial finite element spaces are the nonconforming space of Crouzeix-Raviart for the primal unknown $u$ and the div-conforming space of Raviart-Thomas for the flux $\varphi$ . The two test spaces are the functions constant per cell both for the conservative and for the flux equations. We prove an optimal second order error estimate for the box scheme and we emphasize the link between this scheme and the post-processing of Arnold and Brezzi of the classical mixed method.