2.2 PASS:
Convergence of the Poisson solver with solid boundaries

Author
 Stéphane Popinet
 Command
 sh circle.sh circle.gfs
 Version
 100325
 Required files
 circle.gfs (view) (download)
circle.sh res7.ref error.ref order.ref solution.gfv
 Running time
 3 minutes 51 seconds
Another of the test cases presented in Popinet [31]. A
circular solid boundary of radius 0.25 is embedded in the
domain. The same righthandside is used.
This time the problem does not have an analytical solution and we
use Richardson extrapolation to estimate the error for a given
resolution.
Figure 13: Solution of the Poisson equation. 
Figure 14: Evolution of the residual. 
Figure 15: Average reduction factor. 
Figure 16: Evolution of the error as a function of resolution. 
Figure 17: Corresponding convergence order. 
2.2.1 PASS:
Starshaped solid boundary

Author
 Stéphane Popinet
 Command
 sh ../circle.sh star.gfs
 Version
 100325
 Required files
 star.gfs (view) (download)
res7.ref error.ref order.ref solution.gfv
 Running time
 3 minutes 39 seconds
A similar but more difficult test using a starshaped solid boundary
defined in polar coordinates as
The convergence rate is smaller because the shape of the boundary is
not represented properly on the coarsest levels of the multigrid
hierarchy.
Figure 18: Solution of the Poisson equation. 
Figure 19: Evolution of the residual. 
Figure 20: Average reduction factor. 
Figure 21: Evolution of the error as a function of resolution. 
Figure 22: Corresponding convergence order. 
2.2.2 PASS:
Starshaped solid boundary with refinement

Author
 Stéphane Popinet
 Command
 sh ../circle.sh refined.gfs
 Version
 100325
 Required files
 refined.gfs (view) (download)
res7.ref error.ref order.ref solution.gfv
 Running time
 5 minutes 33 seconds
Same as the star test but with two levels of refinement added near
the solid boundary.
Figure 23: Solution of the Poisson equation. 
Figure 24: Evolution of the residual. 
Figure 25: Average reduction factor. 
Figure 26: Evolution of the error as a function of resolution. 
Figure 27: Corresponding convergence order. 
2.2.3 PASS:
Thin wall at box boundary

Author
 Stéphane Popinet
 Command
 sh ../circle.sh thin.gfs
 Version
 100325
 Required files
 thin.gfs (view) (download)
res7.ref error.ref order.ref solution.gfv
 Running time
 14 minutes 40 seconds
Boxes are used to setup a similar problem but with an infinitely
thin wall.
Figure 28: Solution of the Poisson equation. 
Figure 29: Evolution of the residual. 
Figure 30: Average reduction factor. 
Figure 31: Evolution of the error as a function of resolution. 
Figure 32: Corresponding convergence order. 