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I'm looking at FEM discretizations of $$u_i - \Delta u_i = f$$ for $u_1, u_2$ on subdomains $\Omega_1, \Omega_2$ with interface $\Gamma$. A Neumann-Neumann transmission condition can be formulated by solving for a flux $\lambda$ on $\Gamma$ such that $n\cdot \nabla u_1 = \lambda = -n\cdot \nabla u_2$. One "dual" variational form involves formulating the problem using a Lagrange multiplier unknown $\lambda$, such that

$$\begin{align*} a_1(u_1,v_1) + a_2(u_2,v_2) + \int_{\Gamma} \lambda [[v]]&= (f,v1)+(f,v2)\\ \int_{\Gamma} \mu [[u]] &= 0 \end{align*}$$ and we can eliminate $u_1, u_2$ to solve only in terms of $\lambda$.

Is there a way to do this for other transmission conditions? I can repeat the same process with for Robin interfaces and add $\alpha u_i$ to the interface BCs. Redefining a new Lagrange multiplier $\tilde{\lambda}$ gives $$ \begin{align*} -\alpha u_1 - n\cdot\nabla u_1 &= \tilde{\lambda} = \lambda - \alpha u_1\\ \alpha u_2 - n\cdot\nabla u_2 &= \tilde{\lambda} = -(\lambda - \alpha u_2). \end{align*} $$ but the second BC is no longer coercive (the boundary term contributes negative $\int_{\Gamma}\alpha u_2v_2$ to $a_2(u_2,v_2)$). Is there a way to do a stable Lagrange multiplier formulation for the Robin-Robin case for elliptic problems?

(I'd also be grateful for a reference to previous work if I've overlooked the answer to this in literature.)

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  • $\begingroup$ If no answer appears in a while, it might be good to add a few more definitions to make the question accessible to a wider audience. $\endgroup$ – Abel Molina Jul 8 '14 at 0:34
  • $\begingroup$ I'm not sure there is really an answer - the literature I've seen seems to imply that dual formulations aren't used with Lagrange multipliers. You can update Lagrange multipliers using Robin-Robin transmission conditions, but can't formulate Robin-Robin conditions using Lagrange multipliers. Sorry if it was a bit unaccessible. $\endgroup$ – Jesse Chan Jul 10 '14 at 20:30
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It looks like the answer to the question is no; however, Japhet, Maday and Nataf have come up with a way to formulate Robin transmission conditions in tandem with a Neumann-Neumann Lagrange multiplier approach.

The approach appears to be:

  • initialize $u_0, \lambda_0$ by solving the standard Neumann-Neumann problem with Lagrange multipliers. $\lambda$ represents $\pm n\cdot \nabla u_0$ for Poisson.

  • given $u_0,\lambda_0$ on a given element face, compute $u_1,\lambda_1$ by solving a local problem $$(\nabla u_1,\nabla v_1) + (u_1,v_1) - \int_{\partial K}\lambda_1,v = (f,v)$$ $$\int_{\partial K}(\lambda_1 + \alpha u_1^-)\mu = \int_{\partial K}(\lambda_0 + \alpha u_0^+)\mu$$ where $u_0^+$ is the previous $u$ on the neighboring element, so that information is transmitted across element faces.

  • Repeat until converged.

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