Upper and lower estimates for rate of convergence in the Chernoff product formula for semigroups of operators

Abstract: 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.