[CIG-CS] Gamr status

[Walter Landry]

Greetings Everyone,

The SAMRAI based finite difference Stokes solver is working, so I have
tried out the Sinker benchmark for both.  It turns out that they both
break down even for more moderate viscosity jumps (10^2).  The problem
is that the discretisation can not represent sharp viscosity jumps.
So if you tell the code to keep refining where the error is, the code
will refine all the way to zero cell size.  I have attached pictures
of the pressure in 2D and 3D for both codes.

I am not actually surprised by this result.  Deubelbeiss and Kaus [1]
showed that this is the general result for finite element and finite
difference methods.  The AMR code in Albers [2] and the Rhea code both
did not run into this problem because the viscosity variations are
determined solely by temperature and/or yielding.  Temperature is
always smooth, and it is straightforward to regularize yielding.

So then the question becomes how to handle this.  Suckale et al [3]
got good results using a ghost fluid method with finite differences.
This would require explicit tracking interfaces using something like
level sets.  My main concern is that if there is mixing then
interfaces can become difficult to track and the method will break
down.

There is also regularization.  This may have problems with large
viscosity jumps.  I need to look into this more.

Cheers,
Walter Landry
walter at geodynamics.org

[1] Y. Deubelbeiss and B. Kaus, PEPI 171 (1-4) p. 92-111
[2] Albers, JCP 160, 126$(Q#|(B150 (2000) doi:10.1006/jcph.2000.6438
[3] J. Suckale, J. Nave, B. Hager, JGR 115 B07409,
    doi:10.1029/2009JB006916, 2010
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