I'm following a book about mimetic discretization methods. It states:
Let $v$ e a vector valued function and $f$ a scalar valued function, so $grad \,f $ and $div \, v$ are discretized using convenient grids.
Let $\tilde{v}$ and $\tilde{f}$ be corresponding grid functions to vector spaces of different finite dimensions.
Let $div$, $grad$, $curl$ be the continuous operators and $D$, $G$, $C$ the discretized counterpart.
We enforce that $v = grad \, f$, similarly we also have $v = Gf$
The function $v$ is evaluated at the regular grid points $x_{i} = ih,\; 0 \le i \le n$ and $f$ is evaluated at the staggered grid points $x_{i+\frac{1}{2}} = (i+\frac{1}{2})h,\; 0 \le i \le n-1$ and $h=1/n$.
The discrete operators are stated as:
\begin{align} \left( D\tilde{v} \right)_{i+\frac{1}{2}} &= \frac{v_{i+1} - v_i}{h}, 0 \le i \le n-1 \\ \left( G\tilde{f} \right)_{i} &= \frac{f_{i+\frac{1}{2}} - f_{i-\frac{1}{2}}}{h}, 0 \le i \le n-1 \end{align}
As a boundary condition $f$ is also evaluated at grid boundaries too.
So $\tilde{f} \in \mathbb{R}^{N+2}$, $\tilde{v} \in \mathbb{R}^{N+1}$, $D\tilde{v} \in \mathbb{R}^{N}$ and $G\tilde{f} \in \mathbb{R}^{N+1}$.
I could not understand the reason for this "change" in space after the operators being applied.
In my understand $grad\, f(v_N):\mathbb{R}^N \to \mathbb{R}^N$ and $div \, v_N:\mathbb{R}^N \to \mathbb{R}$.
The relationship:
$$\tilde{v} \in \mathbb{R}^{N+1} = G\tilde{f} \in \mathbb{R}^{N+1}$$
Is clear to me. The biggest issue is about the dimension reduction from $\mathbb{R}^{N+2}$ to $\mathbb{R}^{N+1}$ then $\mathbb{R}^{N}$ in the other relationships.
For reference:
[CASTILLO, MIRANDA] Mimetic discretization methods, CRC Press,2013 pages 43-48
The author states that a $1D$ grid, is formed by a vector that contains elements.
Taken from Mimetic discretization methods, CRC Press,2013 pages 43-48. Copyright by CASTILLO, MIRANDALet's suppose the 1D discretization contains 5 points (0,1,2,3,4). Therefore $N=4$.
The $v$ is evaluated at points (0,1,2,3,4) therefore the vector that represents the $v$ in 1D grid has size 5, hence $v \in \mathbb{R}^{N+1} = \mathbb{R}^5$. The same applies to $Gf$.
$f$ is evaluated at the staggered grid and the boundary conditions, therefore N+2, hence $f \in \mathbb{R}^{N+2}$
$Dv$ is evaluated only at staggered grid points, therefore $Dv \in \mathbb{R}^N$. Same goes to $DGf$.