THE ITERATIVE SOLUTION OF TAYLOR-GALERKIN AUGMENTED MASS MATRIX EQUATIONS

This paper investigates the convergence properties of iterative schemes for the solution of finite element mass matrix equations that arise through the application of a Taylor-Galerkin algorithm to solve instationary Navier-Stokes equations. This is a time-stepping algorithm that involves Galerkin mass matrix equations at fractional stages within each time-step. Plane Poiseuille flow and shear-driven cavity flow are selected as benchmark problems on which to investigate the effects of various choice of scheme and time-step dependency. The iterative convergence of each mass matrix equation for a single fractional stage is studied, both at the element and the system matrix level. The underlying theory is confirmed and it is shown how optimal iterative convergence rates may be achieved for a Jacobi scheme by employing an appropriate acceleration factor. Moreover, this factor is trivial to compute. The consequential effects on the convergence of the time-stepping procedure to reach steady-state are also considered where non-linear effects are present.