[dune-pdelab] Global Integral condition (Lagrange Multiplier) for diffusive problem

[Michael Wenske]

Hello experts,

I want to impose an integral condition upon a diffusive problem in
dune-PDElab.
As an example, assume a box of unity with a circle in the middle
(\Omega_{int}) which is
constrained to u=1. Outside of the circle (\Omega / \Omega_{int}) the
profile is ruled by:

-\Delta u = f(u)  in \Omega / \Omega_{int}, f(u)=u(1-u)

I do not impose any constraints at the outside of the box. Now, without
any further constraints
the only solution to the problem is a profile of u=1 everywhere, due to
the logistic growth
in the reaction term, the flattening by the diffusive operator and the
constraint in the middle.

I want to impose a further constraint to gain a non-trivial solution to
the profile.

\int_{\Omega} u dx = V_{total}

which basically expresses that I am only interested in a solution which
has a certain total
volume in the domain. Please consider the image for visualization.

Now the elegant way (correct me if I'm wrong) to do that would be via a
Lagrange multiplier:

r(u,v) = a(u,v) +l(u,v) + \lambda* (Volume - V_{tot})

Here, r(u,v) is the residuum to be minimized by my newton solver, a(u,v)
is the bilinear form and l(u,v) the
discretisation of the reaction term (as usual).

How would I go about implementing such a constraint in a clean way? The
penalty terms in DG- methods only need local information, so they can be
easily added to the
methods in the local operator. The methods assembling the different
parts of the residuum in the Localoperator only have the local dof's as
arguments but what I need is not a constraint on any single dof, but a
global one
as stated above.

How would I add such a constraint within dune-PDELab? I would also be
thankful if anyone has a hint to
relevant literature for -integral- constraints.

Thanks in advance,

Michael






-------------- next part --------------
A non-text attachment was scrubbed...
Name: 2D_volume_constrained_profile.png
Type: image/png
Size: 33021 bytes
Desc: not available
URL: <https://lists.dune-project.org/pipermail/dune-pdelab/attachments/20190320/3dfecdd4/attachment.png>