10.3 Oseen problems
10.3.2 Polynomial drift
Here, we start with looking at the problem (cf. Definition 7.2.1):
Find u(x, y) :R2 →R2, and p(x, y) :R2 →Rsuch that
−∆u+ (b· ∇)u+cu+∇p=f in Ω∈R2, (10.8)
∇ ·u= 0 in Ω∈R2, (10.9)
with the domain Ω again being the unit square. The velocity field b ist set to b= b1(x, y) We impose the following Dirichlet boundary conditions for (x, y)T ∈∂Ω:
u1(x, y) =
The vector field b depicts a kind of ”drift” through the unit square, its streamlines are shown in Figure 10.21.
Figure 10.21: Streamlines of the convection field
The effect of velocity block strong coupling on the coarsening process
The C/F-splitting algorithm is of course always restricted to the underlying grid. It cannot detect a coarsening direction, when there is no edge into that direction in the according graph imposed by the matrix. Thus, for a vector field b from (10.10), Figure 10.21, where the convection changes its direction slightly from the left border to the right border, the coarse level hierarchy doesn’t give such a clear picture as in the last example.
Figure 10.22: Coarsening structure,ν = 1,θ= 0.6
Figure 10.23: Coarsening structure, ν = 10−6, θ= 0.95
This is especially true for the convection dominated problem in Figure 10.23. However, one could at least observe, that the coarse level structure changes its behaviour from the left to the right half of the domain. In the left half, the coarsening is more along they-direction, and this is shifted a bit more to thex-direction in the right half.
Another difficulty of this flow is the fact, that theamount of convection changes over the domain, even ifν is small. Forν = 10−6 e.g., the local Reynolds number for elements close to (0,0)T is nearly zero, while at the top right corner, it is about 2·106·h. This means, that the problem varies from diffusion dominated to convection dominated over the whole domain, and thus a specific coarsening direction cannot be detected in every part of the domain.
For the diffusion dominated flow however, there is the usual coarsening in all directions, as we see in Figure 10.22, since in this case the convection doesn’t play an important role.
Nevertheless, this problem again corroborates the observation, that rather large coarsening pa-rameters θ (close to 1) are suited for convection dominated problems, while rather moderate values (between 0.5 and 0.7) are required for the diffusion dominated case.
Again, the coarse level properties are also plotted in Figures 10.25 to 10.28 using the legend from Figure 10.24 forν = 1 and 10−2 and Figure 10.7 for ν = 10−4 and 10−6.
theta=0.5 theta=0.6 theta=0.7 theta=0.8 theta=0.9 theta=0.95 theta=0.975
Figure 10.24: Legend for the 2. Oseen problem,ν= 1 and 10−2
h= 1/16 h= 1/32 h= 1/64 h= 1/128 h= 1/256 h= 1/512 θ ravg time ravg time ravg time ravg time ravg time ravg time 0.5 14.5 0.04 13.8 0.08 13.9 0.3 14.0 1.1 13.8 4.9 13.8 21.3 0.6 12.9 0.03 13.5 0.09 13.7 0.3 13.8 1.1 13.6 4.6 13.3 20.0 0.7 11.9 0.03 12.6 0.08 13.1 0.3 13.3 1.1 13.5 4.6 13.8 22.2 0.8 10.9 0.04 12.6 0.09 13.5 0.3 14.2 1.2 14.8 5.0 15.2 24.5 0.9 12.5 0.02 15.3 0.12 18.0 0.5 20.1 2.6 21.5 18.4 22.4 186.1 0.95 11.0 0.03 14.5 0.10 17.6 0.5 20.0 2.4 21.5 18.5 22.4 176.7
Table 10.16: AMG setup times [s] for the Oseen rotation flow,ν = 1,c= 0 h= 1/16 h= 1/32 h= 1/64 h= 1/128 h= 1/256 h= 1/512 θ ravg time ravg time ravg time ravg time ravg time ravg time 0.5 12.3 0.03 13.3 0.08 13.2 0.3 13.6 1.1 12.9 4.4 12.7 19.5 0.6 12.0 0.02 15.0 0.09 13.9 0.3 13.2 1.1 13.1 4.3 13.2 17.4 0.7 11.4 0.02 14.0 0.08 16.6 0.4 13.5 1.1 13.2 4.3 13.1 17.7 0.8 10.3 0.01 12.3 0.07 15.1 0.3 14.3 1.2 13.5 4.4 13.5 19.5 0.9 7.7 0.01 9.8 0.06 11.4 0.3 14.6 1.4 13.9 5.3 14.0 22.2 0.95 6.7 0.02 8.0 0.06 9.3 0.2 11.1 1.0 14.2 6.9 13.7 39.6
Table 10.17: AMG setup times [s] for the Oseen rotation flow, ν= 10−2,c= 0 h= 1/16 h= 1/32 h= 1/64 h= 1/128 h= 1/256 h= 1/512 θ ravg time ravg time ravg time ravg time ravg time ravg time 0.7 9.9 0.01 11.3 0.07 13.2 0.3 15.5 1.8 18.3 15.8 20.7 183.7 0.8 8.5 0.01 10.1 0.06 11.5 0.3 13.5 1.4 15.8 11.8 17.1 107.7 0.9 8.3 0.03 8.5 0.06 9.8 0.2 11.2 1.3 12.7 10.6 12.8 76.1 0.95 7.1 0.02 7.8 0.05 8.6 0.2 9.8 1.1 10.5 6.9 10.8 47.6 0.975 6.3 0.01 7.1 0.04 7.8 0.2 8.5 0.9 9.0 4.4 9.4 30.8 0.9875 6.1 0.02 6.6 0.04 7.1 0.2 7.6 0.8 8.1 4.0 8.4 26.4
Table 10.18: AMG setup times [s] for the Oseen rotation flow, ν= 10−4,c= 0 h= 1/16 h= 1/32 h= 1/64 h= 1/128 h= 1/256 h= 1/512 θ ravg time ravg time ravg time ravg time ravg time ravg time 0.7 9.9 0.02 11.4 0.06 13.2 0.3 14.3 1.6 15.1 12.3 15.9 121.8 0.8 8.8 0.02 10.1 0.07 11.3 0.3 12.2 1.3 12.6 9.0 13.1 88.7 0.9 8.5 0.02 9.1 0.07 9.6 0.2 10.1 1.3 10.5 8.5 10.7 71.1 0.95 7.1 0.02 8.1 0.06 8.5 0.2 8.9 1.0 9.2 6.1 9.4 43.1 0.975 6.7 0.02 7.3 0.04 7.8 0.2 8.0 0.9 8.3 4.5 8.5 28.8 0.9875 6.1 0.02 6.8 0.04 7.3 0.2 7.5 0.8 7.7 4.0 7.8 22.0
Table 10.19: AMG setup times [s] for the Oseen rotation flow, ν= 10−6,c= 0
h= 1/16 h= 1/32 h= 1/64 h= 1/128 h= 1/256 h= 1/512 θ ravg time ravg time ravg time ravg time ravg time ravg time 0.5 14.5 0.03 13.8 0.09 13.9 0.3 14.0 1.1 13.8 4.9 13.8 21.3 0.6 13.1 0.02 13.5 0.09 13.7 0.3 13.8 1.1 13.6 4.6 13.3 19.8 0.7 12.0 0.03 12.6 0.08 13.1 0.3 13.4 1.1 13.6 4.7 13.8 22.0 0.8 10.5 0.03 12.6 0.08 13.5 0.3 14.2 1.2 14.8 5.0 15.2 24.5 0.9 12.5 0.03 15.3 0.11 18.0 0.5 20.1 2.6 21.5 18.3 22.4 185.0 0.95 11.0 0.03 14.5 0.10 17.6 0.5 20.0 2.4 21.5 18.2 22.4 176.5
Table 10.20: AMG setup times [s] for the Oseen rotation flow,ν= 1, c= 1 h= 1/16 h= 1/32 h= 1/64 h= 1/128 h= 1/256 h= 1/512 θ ravg time ravg time ravg time ravg time ravg time ravg time 0.5 12.8 0.03 13.6 0.09 13.5 0.3 13.2 1.1 12.9 4.5 13.0 19.0 0.6 12.6 0.02 15.0 0.09 13.9 0.3 13.0 1.1 13.3 4.3 12.9 17.3 0.7 11.5 0.02 14.2 0.08 17.1 0.4 13.7 1.1 13.5 4.5 12.9 17.6 0.8 10.6 0.02 12.7 0.08 15.2 0.3 14.1 1.2 13.6 4.5 13.6 19.7 0.9 7.8 0.02 9.6 0.07 11.4 0.3 14.5 1.4 13.9 5.5 13.8 22.4 0.95 6.7 0.02 8.2 0.05 9.4 0.2 11.1 1.0 14.2 6.7 13.8 35.9
Table 10.21: AMG setup times [s] for the Oseen rotation flow,ν = 10−2,c= 1 h= 1/16 h= 1/32 h= 1/64 h= 1/128 h= 1/256 h= 1/512 θ ravg time ravg time ravg time ravg time ravg time ravg time 0.7 9.2 0.02 11.1 0.08 13.2 0.3 15.4 1.7 18.4 16.5 20.6 183.8 0.8 8.4 0.01 10.0 0.06 11.2 0.3 13.3 1.4 15.6 11.9 17.1 91.3 0.9 7.7 0.02 8.7 0.06 9.8 0.2 11.1 1.4 12.5 11.1 12.7 69.8 0.95 6.7 0.01 7.9 0.06 8.5 0.2 9.7 1.1 10.4 7.4 10.7 47.6 0.975 6.3 0.01 6.9 0.04 7.7 0.2 8.4 0.9 9.0 5.2 9.4 33.1 0.9875 6.1 0.01 6.7 0.04 7.1 0.2 7.6 0.7 8.1 3.9 8.4 23.2
Table 10.22: AMG setup times [s] for the Oseen rotation flow,ν = 10−4,c= 1 h= 1/16 h= 1/32 h= 1/64 h= 1/128 h= 1/256 h= 1/512 θ ravg time ravg time ravg time ravg time ravg time ravg time 0.7 8.8 0.02 10.8 0.07 12.8 0.3 14.1 1.6 14.9 11.8 16.0 121.8 0.8 8.1 0.02 9.9 0.06 10.9 0.3 12.0 1.2 12.5 8.2 13.1 80.2 0.9 7.6 0.02 8.6 0.06 9.3 0.2 10.0 1.2 10.6 8.1 10.8 64.0 0.95 6.7 0.02 7.5 0.06 8.3 0.2 8.7 1.0 9.2 6.0 9.4 47.6 0.975 6.3 0.02 6.8 0.04 7.5 0.2 7.9 0.8 8.1 4.3 8.5 28.6 0.9875 6.2 0.01 6.6 0.05 7.0 0.2 7.4 0.8 7.6 3.7 7.8 23.1
Table 10.23: AMG setup times [s] for the Oseen rotation flow,ν = 10−6,c= 1
levels
dimensions,nonzeros
0 1 2 3 4
104 105 106
Figure 10.25: Level hierarchy forν = 1
levels
dimensions,nonzeros
0 1 2 3 4
104 105 106
Figure 10.26: Level hierarchy forν = 10−2
levels
dimensions,nonzeros
0 1 2 3 4
104 105 106
Figure 10.27: Level hierarchy forν = 10−4
levels
dimensions,nonzeros
0 1 2 3 4
104 105 106
Figure 10.28: Level hierarchy forν = 10−6
The effect of relaxation
For this Oseen example, we finally would like to examine how the variation of the relaxation parameter influences the convergence behaviour of the V-cycle iteration, as we did for the scalar case.
Thus, for problem (10.8), (10.9) we try out several values forωvel and ωp. Since we didn’t to try every combination, we restricted the pressure relaxation to the set
ωp ∈ {0.1,0.2,0.3}, (10.11)
since we observed, that in the majority of the cases, bigger values resulted in a fast divergence of the method. These values were combined with
ωvel ∈ {0.1,0.2, . . . ,1.8,1.9} (10.12) for the velocity relaxation. Out of these combinations, those values ωp that converged best with the set (10.12) are given in Table 10.28 and the according number of iterations are plotted in the figures in Tables 10.24 to 10.27.
ν = 1 ν = 10−2
ν = 10−4 ν= 10−6
ν = 1 ν = 10−2
ν = 10−4 ν= 10−6
h ν = 1 ν= 10−2 ν= 10−4 ν = 10−6
1/16 0.3 0.3 0.3 0.3
1/32 0.2 0.3 0.3 0.3
1/64 0.1 0.3 0.3 0.3
1/128 0.1 0.3 0.3 0.3
1/256 0.1 0.2 0.2 0.3
1/512 0.1 0.2 0.2 0.2
Table 10.28: Optimal pressure relaxation
First of all, we can state again, that the diffusive problems, i.e. ν = 1 in the left columns of Table 10.24 and 10.26 doesn’t cause bigger problems concerning the choice of ωvel. This is also true independent of the reaction term. Furthermore, the coarsening parameter doesn’t have much influence on the convergence speed (however, it is crucial for the setup, as we have seen in the last section).
The moderate convection dominated case ν = 10−2 in the right columns of Table 10.24 and 10.26 benefits more from a distinct over-relaxation (1.2 ≤ωvel ≤1.8). Also, this problem is more sensible to changes ofθ.
For the convection dominated cases in Tables 10.25 and 10.27 one can observe a stronger tendency towards under-relaxation. However, for ν = 10−6 and decreasing h, the smoothing deteriorates, and the convergence rates heavily slow down.
Convergence speed of the AMG method
In the Figures 10.29 to 10.32, we have exemplarily plotted the convergence histories for a convection-dominated problem atν = 10−4, for the two finest mesh widths h= 1/256 andh = 1/512, with and without reaction term.
time [sec]
residual
0 50 100 150 200
10-11 10-9 10-7 10-5 10-3 10-1 101
AMG
BiCGStab GMRES
Figure 10.29: Convergence of AMG and Krylov methods forν= 10−4,c= 0,h= 2561
time [sec]
residual
0 200 400 600 800
10-11 10-9 10-7 10-5 10-3 10-1 101
AMG
BiCGStab GMRES
Figure 10.30: Convergence of GMRES vs. AMG forν = 10−4,c= 0,h= 5121
time [sec]
Figure 10.31: Convergence of AMG and Krylov methods forν = 10−4,c= 1, h= 2561
Figure 10.32: Convergence of GMRES vs. AMG forν = 10−4,c= 1, h= 5121
Concerning the parameters of the AMG method, we have usedθ= 0.95 for h= 1/256, θ= 0.975 for h = 1/513, c = 1, and θ = 0.9875 for h = 1/513, c = 0. In all cases, we have used the SSOR smoother with ωvel = 1.1,ωvel = 0.2, and two pre- and two post-smoothing steps.
For h = 1/256, five levels were generated, for h = 1/513 we found, that seven generated levels were an optimal compromise between setup-runtime and accuracy of the coarsest level. The Krylov solvers were used with the same parameters as in the last Oseen example.
Compared with the first Oseen problem, we see, that the convergence rates of both AMG and Krylov methods are deteriorated, which is basically due to the strongly varying local Reynolds number.
This complicates the choice of the coarse levels.
Though, we observed, that the AMG coarse level correction still contributes to the error reduction, which is an indicator, that it is not the interpolation that fails. Obviously, in this context, the smoother is the weakest component in the AMG framework.
What we can however state again, is the betterh-scaling of the AMG method. Whileh is getting smaller, the condition number of the matrix increases, which directly degrades the convergence of the Krylov methods, but at the same time increases the gap to AMG.