Finite element approximations for a linear Cahn-Hilliard-Cook equation driven by the space derivativeof a space-time white noise

Georgios T. Kossioris; Georgios E. Zouraris

doi:10.3934/dcdsb.2013.18.1845

Article Contents

2013, Volume 18, Issue 7: 1845-1872. Doi: 10.3934/dcdsb.2013.18.1845

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Finite element approximations for a linear Cahn-Hilliard-Cook equation driven by the space derivative of a space-time white noise

Georgios T. Kossioris^1, and
Georgios E. Zouraris^1,

1.
Department of Mathematics, University of Crete, P.O. Box 2208, GR-710 03 Heraklion, Crete

Received: June 30, 2012

Revised: March 31, 2013

Published: August 2013

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Abstract

We consider an initial- and Dirichlet boundary- value problem for a linear Cahn-Hilliard-Cook equation, in one space dimension, forced by the space derivative of a space-time white noise. First, we propose an approximate stochastic parabolic problem discretizing the noise using linear splines. Then we construct fully-discrete approximations to the solution of the approximate problem using, for the discretization in space, a Galerkin finite element method based on $H^2-$piecewise polynomials, and, for time-stepping, the Backward Euler method. We derive strong a priori estimates: for the error between the solution to the problem and the solution to the approximate problem, and for the numerical approximation error of the solution to the approximate problem.

Keywords:

Finite element method,
space derivative of space-time white noise,
Backward Euler time-stepping,
fully-discrete approximations,
a priori error estimates.

Mathematics Subject Classification: Primary: 65M60, 65M15, 65C20.

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