# The identify with cross product : how to understand?

The problem come from the notations in Hansbo's paper (page 197) and his another paper. Given a domain $\Omega$, the function $u\in H_0^1(\Omega)$. $\Omega$ is divided into 2 subdomains $\Omega_1,\Omega_2$ which are separated by the interface $\Gamma$. The restriction of $u$ on $\Omega_1$ is $u_1$ (i.e., $u|_{\Omega_1}=u_1$) and on $\Omega_2$ is $u_2$ (i.e., $u|_{\Omega_2}=u_2$). He consider the finite element space $V^h=V_1^h\times V^h_2$ where

$V_i^h=\{\phi_i\in H^1(\Omega_i):\phi_i|_{K_i}\text{ is linear },\phi_i|_{\partial \Omega_i}=0\}$.

After that, he constructs the basis functions $\psi$ for space $V^h$ by 2 basis functions $\psi_1,\psi_2$ of $V^h_1,V^h_2$. Then he identify $\psi=(\psi_1,\psi_2)=(\psi|_{\Omega_1},\psi|_{\Omega_2})$.

My question: How can we understand exactly the idea of "identify" in this case? Other words, $\psi=(\psi_1,\psi_2)\in V^h_1\times V^h_2$ whereas $\psi_{\Omega_i}=\psi_i$. For example, take $x\in \Omega_1$, $\psi(x)=\psi_1(x)$ because of the restriction of $\psi$ on $\Omega_1$. However, $\psi(x)$ must be $(\psi_1(x),\psi_2(x))$ because of the identify of $\psi$. It makes me confused!

• Could you summarize the notation here? Sep 25, 2014 at 9:44
• @Travis: sorry, I don't understand your question clearly? Which notations in my question you don't understand? Sep 25, 2014 at 9:47
• The notation seems clear to me now, please disregard the comment. Sep 25, 2014 at 10:53