I have the following problem that the teacher proposed to us in a task. The statement is more or less as follows:
- The object of this problem is the study of a strategy of resolution of type fictitious domain of the following standard problem:
Let $\Omega$ be the unit square of $\mathbb{R}^2$, and $\omega$ a disk strongly included in $\Omega$. We denote $\Gamma$ the border of $\Omega$, and $\gamma$ the border of $\omega$. We give ourselves $f\in L^2(\Omega\setminus \bar{\omega})$, and we will keep the notation $f$ to designate its extension by $0$ over the entire $\Omega$ domain. Then we have $$\begin{cases} -\Delta{u}=f &\text{in $\Omega\setminus \bar{\omega}$} \\ u=0 &\text{on $\Gamma$} \\ u=0& \text{on $\gamma$} \\ \end{cases}$$ The approach aims to develop a method allowing the use of a mesh which does not respect the geometry of inclusion. For the same starting problem, we admit that there is $\tilde{u}\in H^2(\Omega)$ which extends $u$ (initially defined on $\Omega\setminus \bar{\omega}$) all over $\Omega$. For all $g\in L^2(\Omega)$, we note $u_g\in H^2(\Omega)$ the solution of the following auxiliary boundary problem $$\begin{cases} -\Delta{u_g}=f+g &\text{in $\Omega$} \\ u_g=0 &\text{on $\Gamma$} \\ \end{cases}$$ For any function $g\in L^2(\omega)$, we will keep the same notation to designate the function $g\in L^2(\Omega)$ extended by to on the entire domain.
And we also have the following observation: We will take care that if we choose $g = 0$, $u_0$ has no conflict with the solution of the initial problem, because u0 has no reason to cancel on $\gamma$.
We introduce $V=L^2(\omega)$ and $K=\{g\in V,\, u_g|_{\gamma}=0\, \text{a.e on}\, \gamma\}$.
The teacher's questions are:
- Show that $K$ is not empty. We could for example consider $g=-\Delta \tilde{u}|_{\omega}$, where we recall that $\tilde{u}$ is a regular extension of the solution of the initial problem to the whole domain, and show that $g\in K$.
In this part, it was easy for me to show that the function $g$ belongs to $V$ but I'm not very clear on how to show that $u_g|_{\gamma}=0$ on $\gamma$.
- Show that for all $g\in K$, the restriction of $u_g$ to $\Omega$ is identified with the exact solution $u$ of the initial problem.
In this part, I think I should work with the expression $\int_{\Omega\setminus\bar{\omega}} |u_g-u|^2$ and look for a majority type $c \| x-(u_g-u) \|_{H^2(\Omega)}\to 0$ that by $x\to \infty$ and then get that $\int_{\Omega\setminus\bar{\omega}} |u_g-u|^2 = 0$.
I would like to receive some ideas to adequately resolve these two questions and also if you can recommend any pdf document or book in which you could guide me, I would greatly appreciate it.