Convergence of finite volume schemes for semilinear convection diffusion equations

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The topic of this work is the discretization of semilinear elliptic problems in two space dimensions by the cell centered finite volume method. Dirichlet boundary conditions are considered here. A discrete Poincare inequality is used, and estimates on the approximate solutions are proven. The convergence of the scheme without any assumption on the regularity of the exact solution is proven using some compactness results which are shown to hold for the approximate solutions.