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

[Michael Wenske]

Thank you Carsten for answering!

(I take the liberty to forward your reply to the mailing list.)

You are correct, it is a different problem which I solve.

My take on it is, that penalty terms and lagrange multipliers are quite
similar.
In the one case you have a functional f(x) and a condition g(x) = c
which you combine
to the new Lagrange functional to be minimized: L(x,lambda) = f(x) -
lambda*(g(x)-c).
Penalty terms enforcing boundary conditions work in much the same way it
seems.

Anyhow, how would I approach implementing global volume constraints like
this in pdelab?

thanks,

Michael


-------- Forwarded Message --------
Subject: 	Re: [dune-pdelab] Global Integral condition (Lagrange
Multiplier) for diffusive problem
Date: 	Wed, 20 Mar 2019 12:43:39 +0100
From: 	Carsten Gräser <graeser at mi.fu-berlin.de>
To: 	Michael Wenske <m_wens01 at wwu.de>



Hi Michael,
I'm not an expert on PDElab, but I can give some comments
on the outlined approach:

Am 20.03.19 um 12:15 schrieb Michael Wenske:
[...]
> 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.
This cannot work out. If it's correct that there's only one
solution without constraints, then you cannot generate others
by imposing further constraints. You may, however, get other
solutions for an associated minimization problem with constraints,
but this will solve a different PDE.

[...]
> 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})
While I'm not sure what exactly you want to solve, this
does not look like a dual approach with lambda being
a Lagrange multiplier but more like a penalty approach
with penalty factor lambda.

Regards,
Carsten

>
> 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


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