A new approach to the derivation of local extremum diminishing finite element schemes is presented. The monotonicity of an arbitrary Galerkin discretization is enforced by adding discrete diffusion so as to eliminate negative off-diagonal matrix entries. The resulting low-order operator of upwind type acts as a preconditioner within a nonlinear defect correction loop. A multidimensional generalization of TVD concepts is employed to design solution-dependent antidiffusive fluxes which are inserted into the defect vector in order to preclude excessive smearing by numerical diffusion. Standard one-dimensional limiters can be applied edge-by-edge so as to control the slope ratio for the three-point stencil which is reconstructed using a special positivity-preserving gradient recovery. In this paper, a superior limiting strategy is introduced which consists in balancing the diffusive and antidiffusive contributions to each node and applying the resulting correction factors to the incoming antidiffusive fluxes. The proposed algorithm can be readily incorporated into existing flow solvers as a `black-box` postprocessing tool for the matrix assembly routine. Its performance is illustrated by a number of numerical examples for scalar convection problems and incompressible flows in two and three dimensions.