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# Finite element approximation of the deterministic and the stochastic Cahn-Hilliard equation

Ali Mesforush (Institutionen för matematiska vetenskaper, matematik)
Göteborg : Chalmers University of Technology, 2010. ISBN: 978-91-7385-397-2.- 95 s.
[Doktorsavhandling]

This thesis consists of three papers on numerical approximation of the Cahn-Hilliard equation. The main part of the work is concerned with the Cahn-Hilliard equation perturbed by noise, also known as the Cahn-Hilliard-Cook equation. In the ﬁrst paper we consider the linearized Cahn-Hilliard-Cook equation and we discretize it 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. The analysis is set in a framework based on analytic semigroups. The main part of the work consists of detailed error bounds for the corresponding deterministic equation. Time discretization by the implicit Euler method is also considered. In the second paper we study the nonlinear Cahn-Hilliard-Cook equation. 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. In the third paper the deterministic Cahn-Hilliard equation is considered. A posteriori error estimates are proved for a space-time Galerkin ﬁnite element method by using the methodology of dual weighted residuals. We also derive a weight-free a posteriori error estimate in which the weights are condensed into one global stability constant.

Nyckelord: ﬁnite element, a priori error estimate, stochastic integral, mild solution, dual weighted residuals, a posteriori error estimate, additive noise, Wiener process, Cahn-Hilliard equation, existence, regularity, Lya- punov functional, stochastic convolution

Denna post skapades 2010-05-10. Senast ändrad 2016-04-18.
CPL Pubid: 121426

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# Institutioner (Chalmers)

Institutionen för matematiska vetenskaper, matematik (2005-2016)

# Examination

Datum: 2010-06-07
Tid: 10:15
Lokal: Room Pascal, Mathematical Sciences, Chalmers tvärgata 3
Opponent: Dr Matthias Geissert, Technical university, Darmstadt