An algebraic approach to the design of multidimensional high-resolution schemes is introduced and elucidated in the finite element context.  A centered space discretization of unstable
convective terms is rendered local extremum diminishing by a
conservative elimination of negative off-diagonal coefficients from
the discrete transport operator.  This modification leads to an
upwind-biased low-order scheme which is nonoscillatory but overly diffusive. In order to reduce the incurred error, a limited amount of compensating antidiffusion is added in regions where the solution is sufficiently smooth. Two closely related flux correction strategies are presented. The first one is based on a multidimensional
generalization of total variation diminishing (TVD) schemes, whereas
the second one represents an extension of the FEM-FCT paradigm to implicit time-stepping. Nonlinear algebraic systems are solved by an iterative defect correction scheme preconditioned by the low-order evolution operator which enjoys the /mbox{M-matrix} property. The diffusive and antidiffusive terms are represented as a sum of antisymmetric internodal fluxes which are constructed edge-by-edge and
inserted into the global defect vector. The new methodology is applied to scalar transport equations discretized in space by the Galerkin method.Its performance is illustrated by numerical examples for 2D benchmark problems.