A new generalization of the flux-corrected transport (FCT)
methodology to implicit finite element discretizations is proposed.
The underlying high-order scheme is supposed to be unconditionally
stable and produce time-accurate solutions to evolutionary
convection problems. Its nonoscillatory low-order counterpart is
constructed by means of mass lumping followed by elimination of
negative off-diagonal entries from the discrete transport operator.
The raw antidiffusive fluxes, which represent the difference between
the high- and low-order schemes, are updated and limited within an
outer fixed-point iteration. The upper bound for the magnitude of
each antidiffusive flux is evaluated using a single sweep of the
multidimensional FCT limiter at the first outer iteration. This
semi-implicit limiting strategy makes it possible to enforce the
positivity constraint in a very robust and efficient manner.
Moreover, the computation of an intermediate low-order solution can
be avoided. The nonlinear algebraic systems are solved either by a
standard defect correction scheme or by means of a discrete Newton
approach whereby the approximate Jacobian matrix is assembled
edge-by-edge. Numerical examples are presented for two-dimensional
benchmark problems discretized by the standard Galerkin FEM combined
with the Crank-Nicolson time-stepping.