[CIG-CS] Sinker example in Matlab + discretisation errors of the staggered grid FD method for variable viscosity stokes

[Dave May]

Walter,

I want to clarify a couple of comments you've made in relation to the
AMR-StagFD results you have and how they relate to work presented in:
"Discretization errors and free surface stabilization in the finite
difference and marker-in-cell method for applied geodynamics: A
numerical study"

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In response to this:

Subject: [CIG-CS] Gamr status

   "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."

What you mean when you say "breaks down"?
Do you mean to imply that the discretisation fails to converge when
the viscosity jumps is greater than 10^2?
In the work performed in Duretz et. al.,, it was demonstrated that for
several different viscosity structures and contrasts, the StagFD
discretisation is convergent in the L1 norms we define in Eqns
(32,33,34). Note that the important point here is in the norms we
actually used.

   "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."

They use a complete different norm. In Deubelbeiss and Kaus the
L_infinty norm was used. When using StagFD to solve problems with a
discontinuous viscosity structure, you will NOT see convergence in the
L_infinity norm.


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In response to this

Subject: [CIG-CS] Sinker example in Matlab

"However, Duretz et. al.  claimed to get a convergent solution..."
"So I am not sure what is going on with Duretz et. al."

We don't claim we get convergence, we actually demonstrate it.
Again note, that we only demonstrate convergence in a particular norm, i.e. L1
I draw your attention to this line from the paper

"For all our cases, we obtained first‐order convergence in L1 for the
velocity and pressure fields (Table 3)"

In the paper we make no comment about what happens in the L_infinity norm.


I hope this clarifies some of the issues you are having.
In general, discussing convergence is meaningless if you do not
stipulate the norm you are using.


Cheers,
  Dave May