Completion and $S^{-1} A$

Question

Let $\mathfrak{p}$ be a prime ideal of a ring $A$. The completion $\hat{A}$ of $A$ with respect to its adic-topology is used to simplify $A$ beyond the localization $A_{\mathfrak{p}}$. For a multiplicative subset $S \subset A$, we have the localization $S^{-1} A$. Is there an analogous operation for completion? I would like some analogous ring, whose intuition would then be that we are "simplifying $A$ beyond $S^{-1} A$".