  For the case of approximation of convection--diffusion equations using
piecewise affine continuous finite elements a new edge-based nonlinear
diffusion operator is proposed that makes the scheme satisfy a discrete maximum
principle. The diffusion operator is shown to be Lipschitz continuous and
linearity preserving. Using these properties we provide a full stability and
error analysis, which, in the diffusion dominated regime, shows existence,
uniqueness and optimal convergence. Then the algebraic flux correction method
is recalled and we show that the present method can be interpreted as an
algebraic flux correction method for a particular definition of the flux
limiters. The performance of the method is illustrated on some numerical test
cases in two space dimensions.
