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 and what type of object it results in (e.g matrix or vector)? I would really appreciate any help solidifying the main ideas here.
Let $\{ \Omega_i \}$ (for $i = 1,...,m$ ) 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 (statement 2.2 with slight notation change) they state that the global PUM space is given by 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's basis.
I'm having trouble visualizing what would pop out of these sums. In this definition local spaces $V_i$ are matrices whose columns form a basis of the corresponding element of the open cover and partition of unity functions could be piecewise linear shape (hat) functions, for example. Then how would I interpret these sums? Does it give me a single vector? Or a number? Or is it a matrix?
My current thoughts:Each local basis vector $v_i ^{[j]} \in V_i$ has length $N$ (for every local space this length will be the same). The inner sum $\sum _{j=1}^{N_i} \varphi_i v_i ^{[j]}$ is like weighting each local basis vector by the hat function value (each element of the cover has one $\varphi_i(x,y)$). I think $\varphi_i(x,y)$ needs to be evaluated at the same $(x,y)$ values that $v_i ^{[j]}$ is defined on. In which case I think the inner sum is the sum of a $(1,N)$ vector (meaning $\varphi_i(x,y)'$) component wise times each $(N,1)$ vector in the local basis (meaning $v_i ^{[j]}$).This gives us a weighted matrix. And then we use the direct sum to add them up and get a block matrix.
Any help making these ideas more concrete is greatly appreciated. Thanks!