On 27/07/16 11:01, Anna Kalogirou wrote:
> Dear all,
>
> I am trying to solve the system found in the attached pdf as a mixed
> system. I want to solve the same
> problem as the one found here
> <https://bitbucket.org/annakalog/buoy2d/src/14334d3c20b9f10ed7c1246cde9e3cb60b1c75e4/Inequality%20constraint/?at=master>,
> but with the use of Schur complements and not linear algebra.
>
> In a previous discussion about this problem, Lawrence mentioned that the
> test function for integral(lambda*Theta(x-Lp)dx) needs to be considered
> as coming from the real space of constant functions. Could you please
> elaborate on this?

My guess at what Lawrence means is (it's how I would probably approach
it), is that you add an additional equation that says

  (1)  I = integral(lambda*Theta(x-Lp)dx)

and then substitute I in your third equation. Equation (1) can
be turned into a finite element weak formulation by considering the
function space of functions that are constant over the entire domain
This function space is just 1-dimensional and therefore equivalent to
the space of reals R and hence we denote this function space of
constants simply as R. Then we can consider I to be a trial function in
R, and with a test function v4 in R, we can write:

   v4*lambda*Theta(x-Lp)*dx == v4*I*dx/area(domain)

This is a proper finite element equation that you could solve in
conjuction with the other equations. I believe however that this
function space R is not currently implemented in Firedrake. It is
available in fenics, and is on the wishlist.

Cheers
Stephan

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