Analysis of the multigrid method

The analysis of a multigrid method that uses the couple method for solving the incompressible Navier-Stokes equations for the driven cavity and the backward-facing step problems in two and three dimensions. An efficient line solver called symmetrical coupled Gauss-Seidel (SCGS/LS) based upon the SCGS scheme is developed, which acts as a smoother for the nonlinear multigrid method. The Fourier analysis of both SCGS/LS and SCGS are carried out for the two problems which yield lower and upper bounds on the theoretical smoothing factors. Comparing the smoothing capabilities of SCGS/LS and SCGS it is found that the SCGS is not as good in annihilating the high frequency modes at high Reynolds numbers. This is confirmed by numerical experiments using both schemes. The Fourier analysis of both schemes is extended to the three-dimensional lid-driven cavity. Theoretical smoothing factors are obtained by sweeping Fourier modes in the primary flow direction and compared with those obtained numerically. The two-dimensional backward-facing step problem is solved for Reynolds numbers up to 1250. At these values of Reynolds numbers, two secondary separation regions at the opposite sides of the wall are discovered which have not been reported previously. When the SCGS/LS scheme is applied to the three dimensional lid-driven cavity, the results are found to be less efficient than the two dimensional case. This may be due to mismatch in sweeping and flow directions.