The flux-corrected-transport (FCT) paradigm is generalized to implicit
finite element discretizations for hyperbolic systems and applied to
the Euler equations of gas dynamics.

First, the scalar version of the algorithm is presented. Building on
the mathematical theory of positivity-preserving schemes, a
nonoscillatory low-order method is constructed by elimination of
negative off-diagonal matrix entries of the discrete transport
operator. The difference between the linear discretizations of high
and low order is decomposed into sums of internodal antidiffusive
fluxes and embedded into an outer iteration loop of defect correction
type. The monotone low-order operator enjoys the M-matrix property and
constitutes an excellent preconditioner.  Zalesak`s multidimensional
limiter is generalized to implicit time-stepping and used as a
starting point to derive a superior iterative flux limiter independent
of the time step. It is proved that the fully implicit FEM-FCT scheme
is unconditionally positivity-preserving, and a computable upper bound
is derived for the admissible time step under a (semi-)explicit time
discretization.  The methodology is extended to problems with
nonlinear source/sink terms by utilizing a positivity-preserving
linearization technique. Last but not least, the origins of spurious
ripples produced by standard FCT methods in some cases are elucidated
by means of a simple `lever-model` and effective remedies are
proposed.

In the second part of the talk, the scalar FEM-FCT methodology is
carried over to nonlinear hyperbolic systems. An efficient edge-based
algorithm for the global matrix assembly is introduced. The `discrete
upwinding` is performed as in the scalar case by rendering the local
Jacobians situated `off-diagonal´ positive definite. This leads to a
multidimensional extension of Roe`s approximate Riemann solver. For
efficiency reasons, the low-order method can be constructed by adding
scalar dissipation proportional to the spectral radius of the Roe
matrix. The coupled system can be decomposed into scalar subproblems
for individual variables by resorting to a block-diagonal low-order
preconditioner for the outer defect correction loop. However, a
coupled solution strategy based on low-order preconditioning leads to
a better convergence behavior. Application of synchronized correction
factors for an arbitrary set of variables to the conservative fluxes
is shown to preserve positivity.  The performance of the algorithm is
illustrated by numerical examples for the compressible Euler equations
in a wide range of Mach numbers.  Both stationary and time-dependent
results are presented for standard two-dimensional benchmark problems.