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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 res-7.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 right-hand-side 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: Star-shaped solid boundary

Author
Stéphane Popinet
Command
sh ../circle.sh star.gfs
Version
100325
Required files
star.gfs (view) (download)
res-7.ref error.ref order.ref solution.gfv
Running time
3 minutes 39 seconds

A similar but more difficult test using a star-shaped solid boundary defined in polar coordinates as

r(θ)=0.2765+0.079cos(6θ)

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: Star-shaped solid boundary with refinement

Author
Stéphane Popinet
Command
sh ../circle.sh refined.gfs
Version
100325
Required files
refined.gfs (view) (download)
res-7.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)
res-7.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.


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