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

[Walter Landry]

Dave May <dave.mayhem23 at gmail.com> wrote:
> 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$B!>(Border 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.

Does that mean that the L_infinity norm for the inclusion test does
not converge?

This is something that puzzled me for a while.  For many equations, it
is difficult to come up with a scheme that is L1 convergent but not
L_infinity convergent.  That does not seem to be the case for Stokes.
I found this paper

  Kuan-Yu Chen, Ko-An Feng, Yongsam Kim, Ming-Chih Lai, A note on
  pressure accuracy in immersed boundary method for Stokes flow,
  Journal of Computational Physics 230 (2011) 4377$(Q#|(B4383

which found, for a somewhat different method, that the L1 error in the
pressure is first order convergent but the L_infinity error is
constant.

Needless to say, not being able to get the correct solution for the
pressure concerns me.  I could see it having measurable effects with
rheologies that depend on the pressure.  It is one of those things
where you are never really sure that it isn't a problem until you go
back and do it the right way.

Cheers,
Walter Landry
walter at geodynamics.org