2.2.1. Snapshot Space

There are several methods to construct the snapshot space; we will use the harmonic extension of the fine grid functions defined on the boundary of *ωi*. Let us denote *δhl* (*x*) as fine grid delta function, which is defined as *δhl* (*xk*) = *<sup>δ</sup>l*,*<sup>k</sup>* for *xk* ∈ *Jh*(*<sup>ω</sup>i*) where *Jh*(*<sup>ω</sup>i*) denotes the boundary nodes of *ωi*. The snapshot function *ψ<sup>ω</sup><sup>i</sup> l*is then calculated by solving local problem in *ωi*:

$$L(\psi\_l^{\omega\_i}) = 0 \tag{5}$$

subject to the boundary condition *ψ<sup>ω</sup><sup>i</sup> l* = *δhl* (*x*). The snapshot space *V<sup>ω</sup><sup>i</sup> snap* is then constructed as the span of all snapshot functions.
