  Let $\mathscr{I}$ be an ideal sheaf on $P^n$. In the first part of this
paper, we bound the asymptotic regularity of powers of $\mathscr{I}$ as
$ps-3\leq \reg \mathscr{I}^p\leq ps+e$, where $e$ is a constant and $s$ is the
$s$-invariant of $\mathscr{I}$. We also give the same upper bound for the
asymptotic regularity of symbolic powers of $\mathscr{I}$ under some
conditions. In the second part, by using multiplier ideal sheaves, we give a
vanishing theorem of powers of $\mathscr{I}$ when it defines a local complete
intersection subvariety with log canonical singularities.
