[CIG-CS] Variable viscosity Stokes solver

[Wolfgang Bangerth]

> In problems with faulting, we can get arbitrarily small regions with
> arbitrarily large viscosity jumps. [...]

Hm, an interesting case.


> > That actually leads to another interesting question: if the interface is
> > not aligned, what do you use to integrate the local matrices and
> > vectors? If your coefficient, Gauss formula are no longer useful.
> > Presumably an iterated Gauss formula, or something else of low order
> > would be more appropriate.
> 
> Gale partitions the element up with Voronoi cells centered around each
> particle.  The viscosity is assumed smooth on those cells.  So the
> integration is essentially done by using the particles as integration
> points.
> 
> I have also seen good results for other problems by using a higher
> order field (e.g. fourth order) for material properties.  I do not
> know if that will work for these problems.

If you consider Gauss quadrature, it sees the coefficient only in the 
quadrature points. In other words, the end result (i.e. the integral) can't 
distinguish between the original (possibly discontinuous) coefficient and the 
interpolation of this coefficient onto a space of polynomials where the 
interpolation points are equal to the integration points. What this means is 
that we could pretend that the coefficient field has been interpolated to a 
suitably high order polynomial.

The problem with interpolating discontinuous functions is that the 
interpolated version has over- and undershots and may become negative even if 
the original was positive. That makes it desirable to interpolate onto low-
order polynomials (e.g. linears or even constants) on subsets of the cell 
rather than one high-order polynomial on the entire cell. This is what you do 
with the Voronoi subdivision. An alternative is to use iterated low-order 
quadrature formula on subdivisions of the cell.

Best
 W.

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Wolfgang Bangerth                email:            bangerth at math.tamu.edu
                                 www: http://www.math.tamu.edu/~bangerth/