My question concerns Lagrange multipliers and the possibility to impose constraints on the multipliers themselves. I have a Stokes flow which is solved using the Finite Element Method on a domain $\Omega$ having the shape of a square. The equations can be derived as the minimization of energy $\Phi(v\otimes\nabla)$ where $v$ is the velocity subject to the incompressibility condition $v \cdot \nabla = 0$. The pressure $p$ will appear as a Lagrange multiplier. The final weak form of the Stokes problem then takes the form

$$ \int_\Omega \sigma(v\otimes \nabla):[\delta v\otimes \nabla]\mbox{d}\Omega -\int_\Omega p (\delta v \cdot \nabla)\mbox{d}\Omega=0$$ $$ \int_\Omega \delta p (v\cdot \nabla)\mbox{d}\Omega=0$$

Where $\delta v$ and $\delta p$ are testfunctions. Now, I would like the velocity and the pressure to be periodic on the boundaries, i.e $v(x^+)=v(x^-)$ and $p(x^+)=p(x^-)$ where the "+" denotes coordinates on the boundaries of the domain $\Omega$ where the normal is in a positive direction (north and east) and "-" coordinates on the boundary with a negative direction (south and west). I would like to perform this using Lagrange multipliers. The minimization problem can then be stated as

min $\int_\Omega\Phi(v\otimes\nabla)\mbox{d}\Omega$

subject to:

  1. $\nabla\cdot v$=0 on $\Omega$
  2. $v(x^+)-v(x^-)=0$ on $\Gamma=\partial \Omega$
  3. $p(x^+)-p(x^-)=0$ on $\Gamma$

Constraint number 1 will result in a Lagrange multiplier $p$ which is constrained by constraint number 3. As the pressure $p$ already is a Lagrange multiplier, am I allowed to constrain it? I can't see what should be wrong with this, but it is still a quite strange formulation. If this is not at valid approach, how can I solve this problem?

  • $\begingroup$ The usual weak form I've seen has $v \in H(div;\Omega)$, $p \in L^2$. In this case there is no meaning to taking the trace of $p$ along $\Gamma$ since a-priori it may not be continuous. I don't really like this as an answer though, since elliptic regularity implies $p$ will indeed be continuous for sufficiently smooth data. Perhaps have a look in Brezzi and Fortin - Mixed and Hybrid Finite Element Methods if you can get a hold of it..? $\endgroup$ – Nick Alger Apr 19 '12 at 10:04

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