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For the case of approximation of convection--diffusion equations usingpiecewise affine continuous finite elements a new edge-based nonlineardiffusion operator is proposed that makes the scheme satisfy a discrete maximumprinciple. The diffusion operator is shown to be Lipschitz continuous andlinearity preserving. Using these properties we provide a full stability anderror analysis, which, in the diffusion dominated regime, shows existence,uniqueness and optimal convergence. Then the algebraic flux correction methodis recalled and we show that the present method can be interpreted as analgebraic flux correction method for a particular definition of the fluxlimiters. The performance of the method is illustrated on some numerical testcases in two space dimensions.

Edge-based nonlinear diffusion for finite element approximations of  convection-diffusion equations and its relation to algebraic flux-correction  schemes