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.