  Chernoff approximations to strongly continuous one-parameter semigroups give
solutions to a wide class of differential equations. This paper studies the
rate of convergence of the Chernoff approximations. We provide simple natural
examples for which the convergence is arbitrary fast, is arbitrary slow, and
holds in the strong operator topology but does not hold in the norm operator
topology. We also prove a general theorem that gives an upper estimate for the
speed of decay of the norm of the residual term of the Chernoff approximations.
The result is applied to one-dimensional parabolic differential equations with
variable coefficients. The obtained estimates can be used for the numerical
solution of PDEs.
