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<p>Thank you Carsten for answering!<br>
</p>
<p>(I take the liberty to forward your reply to the mailing list.)</p>
<p>You are correct, it is a different problem which I solve.<br>
<br>
My take on it is, that penalty terms and lagrange multipliers are
quite similar.<br>
In the one case you have a functional f(x) and a condition g(x) =
c which you combine<br>
to the new Lagrange functional to be minimized: L(x,lambda) = f(x)
- lambda*(g(x)-c).<br>
Penalty terms enforcing boundary conditions work in much the same
way it seems.<br>
<br>
Anyhow, how would I approach implementing global volume
constraints like this in pdelab?</p>
<p>thanks,</p>
<p>Michael<br>
</p>
<div class="moz-forward-container"><br>
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<th valign="BASELINE" nowrap="nowrap" align="RIGHT">Subject:
</th>
<td>Re: [dune-pdelab] Global Integral condition (Lagrange
Multiplier) for diffusive problem</td>
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<th valign="BASELINE" nowrap="nowrap" align="RIGHT">Date: </th>
<td>Wed, 20 Mar 2019 12:43:39 +0100</td>
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<th valign="BASELINE" nowrap="nowrap" align="RIGHT">From: </th>
<td>Carsten Gräser <a class="moz-txt-link-rfc2396E" href="mailto:graeser@mi.fu-berlin.de"><graeser@mi.fu-berlin.de></a></td>
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<th valign="BASELINE" nowrap="nowrap" align="RIGHT">To: </th>
<td>Michael Wenske <a class="moz-txt-link-rfc2396E" href="mailto:m_wens01@wwu.de"><m_wens01@wwu.de></a></td>
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<br>
<br>
Hi Michael,<br>
I'm not an expert on PDElab, but I can give some comments<br>
on the outlined approach:<br>
<br>
Am 20.03.19 um 12:15 schrieb Michael Wenske:<br>
[...]<br>
<blockquote type="cite">I do not impose any constraints at the
outside of the box. Now, without<br>
any further constraints<br>
the only solution to the problem is a profile of u=1 everywhere,
due to<br>
the logistic growth<br>
in the reaction term, the flattening by the diffusive operator
and the<br>
constraint in the middle.<br>
<br>
I want to impose a further constraint to gain a non-trivial
solution to<br>
the profile.<br>
</blockquote>
This cannot work out. If it's correct that there's only one<br>
solution without constraints, then you cannot generate others<br>
by imposing further constraints. You may, however, get other<br>
solutions for an associated minimization problem with constraints,<br>
but this will solve a different PDE.<br>
<br>
[...]<br>
<blockquote type="cite">Now the elegant way (correct me if I'm
wrong) to do that would be via a<br>
Lagrange multiplier:<br>
<br>
r(u,v) = a(u,v) +l(u,v) + \lambda* (Volume - V_{tot})<br>
</blockquote>
While I'm not sure what exactly you want to solve, this<br>
does not look like a dual approach with lambda being<br>
a Lagrange multiplier but more like a penalty approach<br>
with penalty factor lambda.<br>
<br>
Regards,<br>
Carsten<br>
<br>
<blockquote type="cite"><br>
Here, r(u,v) is the residuum to be minimized by my newton
solver, a(u,v)<br>
is the bilinear form and l(u,v) the<br>
discretisation of the reaction term (as usual).<br>
<br>
How would I go about implementing such a constraint in a clean
way? The<br>
penalty terms in DG- methods only need local information, so
they can be<br>
easily added to the<br>
methods in the local operator. The methods assembling the
different<br>
parts of the residuum in the Localoperator only have the local
dof's as<br>
arguments but what I need is not a constraint on any single dof,
but a<br>
global one<br>
as stated above.<br>
<br>
How would I add such a constraint within dune-PDELab? I would
also be<br>
thankful if anyone has a hint to<br>
relevant literature for -integral- constraints.<br>
<br>
Thanks in advance,<br>
<br>
Michael<br>
</blockquote>
<br>
<br>
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