Tools for simulating non‐stationary incompressible flow via discretely divergence‐free finite element models

We develop simulation tools for the non‐stationary incompressible 2D Navier‐‐Stokes equations. The most important components of the finite element code are: the fractional step ϑ‐scheme, which is of second‐order accuracy and strongly A‐stable, for the time discretization; a fixed point defect correction method with adaptive step length control for the non‐linear problems (stationary Navier‐Stokes equations); a modified upwind discretization of higher‐order accuracy for the convective terms. Finally, the resulting nonsymmetric linear subproblems are treated by a special multigrid algorithm which is adapted to the quadrilateral non‐conforming discretely divergence‐free finite elements. For the graphical postprocess we use a fully non‐stationary and interactive particle‐tracing method. With extensive test calculations we show that our method is a candidate for a ‘black box’ solver.