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We prove here convergence results for the approximate finite volume solutions of a diffusion problem with mixed Dirichlet, Neumann and Signorini boundary conditions which is formulated as a variational inequality. The convergence result is also shown to be easily adapted to the case of the obstacle problem which also writes as a variational inequality. An error estimate of order one with respect to the mesh size is given when the solutions to the continuous problems belong to H 2 (Ω).doi:10.1093/imanum/21.2.553 fatcat:et6dq2q2ezej3a6vd6gtwrfz4u