The flux-corrected transport (FCT) methodology is generalized to
implicit finite element schemes and applied to the Euler equations
of gas dynamics. For scalar equations, a local extremum diminishing
scheme is constructed by adding artificial diffusion so as to
eliminate negative off-diagonal entries from the high-order
transport operator. To obtain a nonoscillatory low-order method in
the case of hyperbolic systems, the artificial viscosity tensor is
designed so that all off-diagonal blocks of the discrete Jacobians
are rendered positive semi-definite. Compensating antidiffusion is
applied within a fixed-point defect correction loop so as to recover
the high accuracy of the Galerkin discretization in regions of
smooth solutions. All conservative matrix manipulations are
performed edge-by-edge which leads to an efficient algorithm for the
matrix assembly.