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 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 finite element method combined with the Crank–Nicolson time stepping. Copyright © 2007 John Wiley & Sons, Ltd.