Consider a Poisson boundary value problem on a domain $\Omega$ with non-overlapping Dirichlet and Neumann boundary segments that construct the domain boundary $\partial \Omega = \Gamma_N \cup \Gamma_D$. \begin{align} -\nabla^2 u &= f\quad\ \ \ \text{on} \ \Omega \\ u &= u_D \quad \text{on}\ \Gamma_D \\ -\partial u/\partial n &= g_N \quad \text{on}\ \Gamma_N \end{align} with the weak form

$$\int_{\Omega} \nabla u \cdot \nabla v dx - \int_{\Gamma_D} \nabla u \cdot\textbf{n} v ds = \int_{\Omega} fv dx + \int_{\Gamma_N} g_Nv ds$$

Note that the second term on the LHS vanishes if we request the test function $v$ to belong to $H_0^1(\Omega)$. Yet, (spoiler alert :) ) since my post exactly questions that I decided to keep it for the time being.

As far as I understood, the weak imposition of Dirichlet BC requires the first BC to hold only weakly, i.e., for all test functions $\mu$ in a proper space $Q$ we must have

$$\int_{\Gamma_D}(u-u_D) \mu ds = 0 \hspace{1cm} \forall \mu \in Q$$ Thus the weak form will take the form of a mixed problem:

$$\int_{\Omega} \nabla u \cdot \nabla v dx + \int_{\Gamma_D} lv ds = \int_{\Omega} fv dx + \int_{\Gamma_N} g_Nv ds$$ $$\int_{\Gamma_D}u \mu ds = \int_{\Gamma_D}u_D \mu ds$$ for trial functions $(u,l)=:(u,-\partial u/ \partial n)$ and test functions $(v, \mu)$.

My question is what are the proper spaces for $u,v, l$, and $\mu$?

I'm only sure that:

  • $u$ doesn't need to be in $H_{u_D}^1(\Omega)$ (otherwise, the second equation would be trivial.)
  • $v$ isn't in $H_0^1(\Omega)$ (otherwise, the second term in the LHS of the first equation would vanish, leading to a mixed problem with one test function and two trials - which might make sense, but never have seen anything like that...)
  • $\begingroup$ Normally you would strongly impose the Dirichlet BC. Why would you impose it weakly? $\endgroup$ – Ian Jul 21 at 21:42
  • $\begingroup$ @Ian I am not sure myself but I can find many papers that consider such formulation for modelling fluid flows. $\endgroup$ – Korf Jul 22 at 11:31
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    $\begingroup$ Normally you would do this when there isn't enough regularity to make sense of the trace, but there is here. Is this just a toy model for testing a numerical method meant for another kind of problem? $\endgroup$ – Ian Jul 22 at 11:48
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    $\begingroup$ @Ian well, I'm trying to build intuition around this formulation, and the Poisson problem is just the simplest example. But it would be interesting to use it as a toy problem against strong imposition as well. $\endgroup$ – arash Jul 22 at 14:55
  • $\begingroup$ OK. Well, a question like this makes a bit more sense if you clarify what kind of regularity you expect to be dealing with. Because again, here it doesn't really make sense to do things this way, because you have enough regularity that the trace is defined even if $f$ is just $L^2$. $\endgroup$ – Ian Jul 22 at 14:56

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