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Abstract. We introduce and analyse a finite volume method for the discretization of elliptic boundary value problems in R 2 . The method is based on nonuniform triangulations with piecewise linear nonconforming spaces. We prove optimal order error estimates in the L 2 -norm and a mesh dependent H 1 -norm.

Abstract. We consider a new formulation for finite volume element methods, which is satisfied by known finite volume methods and it can be used to introduce new ones. This framework results by approximating the test function in the formulation of finite element method. We analyze piecewise linear conforming or nonconforming approximations on nonuniform triangulations and prove optimal order H 1 −norm and L 2 −norm error estimates.Mathematics Subject Classification. 65N30, 65N15.

We analyze the spatially semidiscrete piecewise linear finite volume element method for parabolic equations in a convex polygonal domain in the plane. Our approach is based on the properties of the standard finite element Ritz projection and also of the elliptic projection defined by the bilinear form associated with the variational formulation of the finite volume element method. Because the domain is polygonal, special attention has to be paid to the limited regularity of the exact solution. We give sufficient conditions in terms of data that yield optimal order error estimates in L 2 and H 1 . The convergence rate in the L ϱ norm is suboptimal, the same as in the corresponding finite element method, and almost optimal away from the corners. We also briefly consider the lumped mass modification and the backward Euler fully discrete method.