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

[Carsten Gräser]

Am 20.03.19 um 13:36 schrieb 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.
Unfortunately they are not:

> 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).
Yes, that's a classical formulation with a Lagrange multiplier.
Here you would look for saddle point of L which is a (x,lambda)
pair. In the penalty formulation you would look for a minimizer x of

  J(x) = f(x) + lambda*|g(x)-c|^2

for fixed scalar lambda. This essentially corresponds to your nonlinear
residual r(.,.).

> 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?
I think it would be helpful to know which of those formulations you aim
at, since they are _very_ different.

Best,
Carsten

> 
> 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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-- 
Prof. Dr. Carsten Gräser
Freie Universität Berlin
Institut für Mathematik
Arnimallee 6
14195 Berlin, Germany
phone: +49 30 838 72637
fax  : +49 30 838 472637
email: graeser at mi.fu-berlin.de
URL  : http://page.mi.fu-berlin.de/graeser

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