Finite Element Approximation of the Cahn–Hilliard–Cook Equation

Abstract

Abstract. We study the nonlinear stochastic Cahn-Hilliard equation perturbed by additive colored noise. We show almost sure existence and regularity of solutions. We introduce spatial approximation by a standard finite element method and prove error estimates of optimal order on sets of probability arbitrarily close to 1. We also prove strong convergence without known rate.

Erratum: Finite Element Approximation of the Cahn--Hilliard--Cook Equation

“…Theorem 2.1 provides optimal order of convergence for the linearized Cahn-Hilliard-Cook equation in a stronger topology than the one in [3, Theorem 2.2] in exchange for a slight additional regularity requirement on the covariance operator Q. In particular, it implies pathwise convergence with essentially optimal rate for the linearized equation, which is an important ingredient in the proof of the main result in [3].…”

“…The main gap in [3] occurs when deriving the last inequality on page 2426 using [3, Theorem 2.2]. Indeed, one could then only conclude the existence of a set Ω = Ω ,h,t such that the inequality holds.…”

“…Indeed, one could then only conclude the existence of a set Ω = Ω ,h,t such that the inequality holds. The dependence on t of the set then compromises the rest of the proof of [3,Theorem 5.3] and hence also the proof of [3,Theorem 5.4]. This can be avoided by using Theorem 2.1 instead.…”

“…The proof of [3,Theorem 5.3] is incomplete and in the present note we provide an additional convergence result to fill in the gap. In order to do so one has to replace [3, Theorem 2.2], which is quoted from [4], by Theorem 2.1 below.…”

“…In Section 2 we state and prove the result which is missing from [3] and in Section 3 we outline what additional small changes one has to make in the arguments of [3] as a consequence.…”