I'm working to understand how the partition of unity method is used to build a global PUM space from local approximation spaces. Can someone please explain the mechanics of gluing the local spaces together? I would really appreciate any examples or help demystifying the main ideas here.
Let $\{ \Omega_i \}$ be an open over of $\Omega \subset \mathbb{R}^n$ and let $\{ \varphi_i\}$ be a $(M,C_{\infty},C_G)$ partition of unity subordinate to $\{ \Omega_i \}$. Let $V_i \subset H^1(\Omega_i \cap\Omega)$ (the local approximation spaces) be given. In this paper they state that the trial and test spaces for the generalized finite element method are given by the PUM space which is constructed from the local approximation spaces and they give the definition:$ S(\Omega) = \{\psi : \psi = \sum _{i = 1} ^m \sum _{j=1}^{N_i} \varphi_i v_i ^{[j]} \mbox{ where } v_i ^{[j]} \in V_i \}$ Note here that $N_i$ is the size of each local approximation space and $m$ is the total number of subdomains for the whole global space.
I'm trying to make these ideas more concrete because I'm having trouble visualizing what would pop out of these sums. Say my $V_i$ are matrices whose columns are basis functions of my local approximation spaces and my partition of unity functions are piecewise linear shape (hat) functions for example. Then how would I interpret these sums? Does it give me a single vector? Or is it a matrix? The hat functions are overlapping so it makes sense that we need to add all the subdomains so we get all contributions over all areas but I'm still confused on how this gives me a matrix I can use for GFEM.
Any help making these ideas more concrete is greatly appreciated. Thanks!