Implicit FEM-FCT algorithms and discrete Newton methods for transient convection problems

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 highand low-order schemes, are updated and limited within an outer fixedpoint 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.