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Finite-element approximation of the linearized Cahn-Hilliard-Cook equation

Journal article
Authors Stig Larsson
Ali Mesforush
Published in IMA Journal of Numerical Analysis
Volume 31
Issue 4
Pages 1315-1333
ISSN 0272-4979
Publication year 2011
Published at Department of Mathematical Sciences, Mathematics
Pages 1315-1333
Language en
Links dx.doi.org/10.1093/imanum/drq042
https://gup.ub.gu.se/file/66279
Keywords Cahn–Hilliard–Cook equation, stochastic convolution, Wiener process, finite-element method, backward Euler method, mean square error estimate, strong convergence
Subject categories Numerical analysis

Abstract

The linearized Cahn–Hilliard–Cook equation is discretized in the spatial variables by a standard finite-element method. Strong convergence estimates are proved under suitable assumptions on the covariance operator of the Wiener process, which is driving the equation. Backward Euler time stepping is also studied. The analysis is set in a framework based on analytic semigroups. The main effort is spent on proving detailed error bounds for the corresponding deterministic Cahn–Hilliard equation. The results should be interpreted as results on the approximation of the stochastic convolution, which is a part of the mild solution of the nonlinear Cahn–Hilliard–Cook equation.

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