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Discrete and Continuous Dynamical Systems - Series B (DCDS-B)
 

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

Pages: 1845 - 1872, Volume 18, Issue 7, September 2013      doi:10.3934/dcdsb.2013.18.1845

 
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Georgios T. Kossioris - Department of Mathematics, University of Crete, P.O. Box 2208, GR-710 03 Heraklion, Crete, Greece (email)
Georgios E. Zouraris - Department of Mathematics, University of Crete, P.O. Box 2208, GR-710 03 Heraklion, Crete, Greece (email)

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.

Received: July 2012;      Revised: April 2013;      Available Online: May 2013.

 References