Let $X$ be a scheme, $D$ an effective divisor on $X$ with structure sheaf $\mathcal{O}_D$, and $U = X\setminus D$. I think I need $D$ either ample or affine. If necessary we can assume $\mathcal{O}_X$ acyclic or other niceness conditions on $X$.
The global sections $H^0(X,\mathcal{O}_D(nD))$ should all be isomorphic, and I am trying to describe the isomorphism in as natural a way as possible, for example along the lines of the the local isomorphisms $\mathcal{O}_X|_U \xrightarrow{\times f^{-n}} \mathcal{O}_X(nD)|_U$, where $f$ is an equation defining $D$.
More generally I expect the same should hold for the $H^0\left(X,\mathcal{F}\otimes\mathcal{O}_D(nD)\right)$, for another locally free sheaf $\mathcal{F}$: $$H^0\left(X,\mathcal{F}\otimes\mathcal{O}_D\right) \xrightarrow{\sim}H^0\left(X,\mathcal{F}\otimes\mathcal{O}_D(nD)\right)$$
Question: What is the most natural way of exhibiting such an isomorphism?
Assume $N\geq0$ is such that $H^1\left(X,\mathcal{O}_X\big((N-1)D\big)\right) = 0$, then $$0 \to \mathcal{O}_X\left((N-1)D\right) \to \mathcal{O}_X(ND) \to \mathcal{O}_D(ND) \to 0$$ descends to an exact sequence $$0 \to H^0\left(X,\mathcal{O}_X\big((N-1)D\big)\right) \to H^0\left(X,\mathcal{O}_X(ND)\right) \to H^0\left(\mathcal{O}_D(ND)\right) \to 0$$ which indicates that the $\mathcal{O}_D(ND)$ is capturing all the global sections of $\mathcal{O}_X(ND)$ that are not contained in $\mathcal{O}_X\big((N-1)D\big)$.
Ultimately, for some other locally free sheaf $\mathcal{F}$, I would like to describe a decomposition of $H^0\left(X, \mathcal{F}(nD)\right)$ in terms of the $H^0(X,\mathcal{F}\otimes\mathcal{O}_D(nD))$.
This leads me to my second
Question: When can I obtain a natural injection $$H^0(X, \mathcal{O}_X(nD) \hookrightarrow \bigoplus\limits_{n\geq 0} H^0\left(X, \mathcal{O}_D(nD)\right)$$ into the ring of sections of the "conormal cone" of $D$, or more generally $$H^0(X, \mathcal{F}_X(nD) \hookrightarrow \bigoplus\limits_{n\geq 0} H^0\left(X, \mathcal{F}\otimes\mathcal{O}_D(nD)\right)?$$
This has worked in the simple concrete examples I have been looking at, but I'm not sure how to prove it in general. Any help would be much appreciated.