# American Institute of Mathematical Sciences

September  2013, 18(7): 1845-1872. doi: 10.3934/dcdsb.2013.18.1845

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

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

Received  July 2012 Revised  April 2013 Published  May 2013

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.
Citation: Georgios T. Kossioris, Georgios E. Zouraris. Finite element approximations for a linear Cahn-Hilliard-Cook equation driven by the space derivative of a space-time white noise. Discrete and Continuous Dynamical Systems - B, 2013, 18 (7) : 1845-1872. doi: 10.3934/dcdsb.2013.18.1845
##### References:
 [1] E. J. Allen, S. J. Novosel and Z. Zhang, Finite element and difference approximation of some linear stochastic partial differential equations, Stochastics Stochastics Rep., 64 (1998), 117-142. doi: 10.1080/17442509808834159. [2] A. Are, M. A. Katsoulakis and A. Szepessy, Coarse-grained Langevin approximations and spatiotemporal acceleration for kinetic Monte Carlo simulations of diffusion of interacting particles, Chin. Ann. Math. Series B, 30 (2009), 653-682. doi: 10.1007/s11401-009-0219-x. [3] L. Bin, "Numerical Method for a Parabolic Stochastic Partial Differential Equation," Master Thesis 2004-03, Chalmers University of Technology, Göteborg, Sweden, 2004. [4] D. Blömker, S. Maier-Paape and T. Wanner, Second phase spinonal decomposition for the Cahn-Hilliard-Cook equation, Transactions of the AMS, 360 (2008), 449-489. doi: 10.1090/S0002-9947-07-04387-5. [5] J. H. Bramble and S. R. Hilbert, Estimation of linear functionals on Sobolev spaces with application to Fourier transforms and spline interpolation, SIAM J. Numer. Anal., 7 (1970), 112-124. doi: 10.1137/0707006. [6] A. Debussche and L. Zambotti, Conservative stochastic Cahn-Hilliard equation with reflection, Annals of Probability, 35 (2007), 1706-1739. doi: 10.1214/009117906000000773. [7] N. Dunford and J. T. Schwartz, "Linear Operators. Part II. Spectral Theory. Self Adjoint Operators in Hilbert Space," Reprint of the 1963 original, Wiley Classics Library, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1988. [8] W. Grecksch and P. E. Kloeden, Time-discretised Galerkin approximations of parabolic stochastic PDE's, Bull. Austral. Math. Soc., 54 (1996), 79-85. doi: 10.1017/S0004972700015094. [9] G. H. Golub and C. F. Van Loan, "Matrix Computations," Second Edition, The John Hopkins University Press, Baltimore, 1989. [10] P. C. Hohenberg and B. I. Halperin, Theory of dynamic critical phenomena, J. Rev. Mod. Phys., 49 (1977), 435-479. doi: 10.1103/RevModPhys.49.435. [11] G. Kallianpur and J. Xiong, "Stochastic Differential Equations in Infinite Dimensional Spaces," Institute of Mathematical Statistics, Lecture Notes-Monograph Series 26, Hayward, California, 1995. [12] M. A Katsoulakis and D. G. Vlachos, Coarse-grained stochastic processes and kinetic Monte Carlo simulators for the diffusion of interacting particles, J. Chem. Phys., 119 (2003), 9412-9427. doi: 10.1063/1.1616513. [13] P. E. Kloeden and S. Shot, Linear-implicit strong schemes for Itô-Galerkin approximations of stochastic PDE's, Journal of Applied Mathematics and Stochastic Analysis, 14 (2001), 47-53. doi: 10.1155/S1048953301000053. [14] G. T. Kossioris and G. E. Zouraris, Fully-discrete finite element approximations for a fourth-order linear stochastic parabolic equation with additive space-time white noise, Mathematical Modelling and Numerical Analysis, 44 (2010), 289-322. doi: 10.1051/m2an/2010003. [15] G. T. Kossioris and G. E. Zouraris, Finite element approximations for a linear fourth-order parabolic SPDE in two and three space dimensions with additive space-time white noise, Applied Numerical Mathematics, 67 (2013), 243-261. doi: 10.1016/j.apnum.2012.01.003. [16] S. Larsson and A. Mesforush, Finite element approximation of the linearized Cahn-Hilliard-Cook equation, IMA J. Numer. Anal., 31 (2011), 1315-1333. doi: 10.1093/imanum/drq042. [17] J. L. Lions and E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications. Vol. I," Springer-Verlag, Berlin - Heidelberg, 1972. [18] J. Printems, On the discretization in time of parabolic stochastic partial differential equations, Mathematical Modelling and Numerical Analysis, 35 (2001), 1055-1078. doi: 10.1051/m2an:2001148. [19] T. M. Rogers, K. R. Elder and R. C. Desai, Numerical study of the late stages of spinodal decomposition, Physical Review B, 37 (1988), 9638-9649. doi: 10.1103/PhysRevB.37.9638. [20] V. Thomée, "Galerkin Finite Element Methods for Parabolic Problems," Spriger Series in Computational Mathematics, vol. 25, Springer-Verlag, Berlin Heidelberg, 1997. [21] Y. Yan, Galerkin finite element methods for stochastic parabolic partial differential equations, SIAM Journal on Numerical Analysis, 43 (2005), 1363-1384. doi: 10.1137/040605278. [22] J. B. Walsh, "An Introduction to Stochastic Partial Differential Equations," in "Lecture Notes in Mathematics no. 1180", Springer Verlag, 1986, 265-439. doi: 10.1007/BFb0074920. [23] J. B. Walsh, Finite element methods for parabolic stochastic PDE's, Potential Analysis, 23 (2005), 1-43. doi: 10.1007/s11118-004-2950-y.

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##### References:
 [1] E. J. Allen, S. J. Novosel and Z. Zhang, Finite element and difference approximation of some linear stochastic partial differential equations, Stochastics Stochastics Rep., 64 (1998), 117-142. doi: 10.1080/17442509808834159. [2] A. Are, M. A. Katsoulakis and A. Szepessy, Coarse-grained Langevin approximations and spatiotemporal acceleration for kinetic Monte Carlo simulations of diffusion of interacting particles, Chin. Ann. Math. Series B, 30 (2009), 653-682. doi: 10.1007/s11401-009-0219-x. [3] L. Bin, "Numerical Method for a Parabolic Stochastic Partial Differential Equation," Master Thesis 2004-03, Chalmers University of Technology, Göteborg, Sweden, 2004. [4] D. Blömker, S. Maier-Paape and T. Wanner, Second phase spinonal decomposition for the Cahn-Hilliard-Cook equation, Transactions of the AMS, 360 (2008), 449-489. doi: 10.1090/S0002-9947-07-04387-5. [5] J. H. Bramble and S. R. Hilbert, Estimation of linear functionals on Sobolev spaces with application to Fourier transforms and spline interpolation, SIAM J. Numer. Anal., 7 (1970), 112-124. doi: 10.1137/0707006. [6] A. Debussche and L. Zambotti, Conservative stochastic Cahn-Hilliard equation with reflection, Annals of Probability, 35 (2007), 1706-1739. doi: 10.1214/009117906000000773. [7] N. Dunford and J. T. Schwartz, "Linear Operators. Part II. Spectral Theory. Self Adjoint Operators in Hilbert Space," Reprint of the 1963 original, Wiley Classics Library, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1988. [8] W. Grecksch and P. E. Kloeden, Time-discretised Galerkin approximations of parabolic stochastic PDE's, Bull. Austral. Math. Soc., 54 (1996), 79-85. doi: 10.1017/S0004972700015094. [9] G. H. Golub and C. F. Van Loan, "Matrix Computations," Second Edition, The John Hopkins University Press, Baltimore, 1989. [10] P. C. Hohenberg and B. I. Halperin, Theory of dynamic critical phenomena, J. Rev. Mod. Phys., 49 (1977), 435-479. doi: 10.1103/RevModPhys.49.435. [11] G. Kallianpur and J. Xiong, "Stochastic Differential Equations in Infinite Dimensional Spaces," Institute of Mathematical Statistics, Lecture Notes-Monograph Series 26, Hayward, California, 1995. [12] M. A Katsoulakis and D. G. Vlachos, Coarse-grained stochastic processes and kinetic Monte Carlo simulators for the diffusion of interacting particles, J. Chem. Phys., 119 (2003), 9412-9427. doi: 10.1063/1.1616513. [13] P. E. Kloeden and S. Shot, Linear-implicit strong schemes for Itô-Galerkin approximations of stochastic PDE's, Journal of Applied Mathematics and Stochastic Analysis, 14 (2001), 47-53. doi: 10.1155/S1048953301000053. [14] G. T. Kossioris and G. E. Zouraris, Fully-discrete finite element approximations for a fourth-order linear stochastic parabolic equation with additive space-time white noise, Mathematical Modelling and Numerical Analysis, 44 (2010), 289-322. doi: 10.1051/m2an/2010003. [15] G. T. Kossioris and G. E. Zouraris, Finite element approximations for a linear fourth-order parabolic SPDE in two and three space dimensions with additive space-time white noise, Applied Numerical Mathematics, 67 (2013), 243-261. doi: 10.1016/j.apnum.2012.01.003. [16] S. Larsson and A. Mesforush, Finite element approximation of the linearized Cahn-Hilliard-Cook equation, IMA J. Numer. Anal., 31 (2011), 1315-1333. doi: 10.1093/imanum/drq042. [17] J. L. Lions and E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications. Vol. I," Springer-Verlag, Berlin - Heidelberg, 1972. [18] J. Printems, On the discretization in time of parabolic stochastic partial differential equations, Mathematical Modelling and Numerical Analysis, 35 (2001), 1055-1078. doi: 10.1051/m2an:2001148. [19] T. M. Rogers, K. R. Elder and R. C. Desai, Numerical study of the late stages of spinodal decomposition, Physical Review B, 37 (1988), 9638-9649. doi: 10.1103/PhysRevB.37.9638. [20] V. Thomée, "Galerkin Finite Element Methods for Parabolic Problems," Spriger Series in Computational Mathematics, vol. 25, Springer-Verlag, Berlin Heidelberg, 1997. [21] Y. Yan, Galerkin finite element methods for stochastic parabolic partial differential equations, SIAM Journal on Numerical Analysis, 43 (2005), 1363-1384. doi: 10.1137/040605278. [22] J. B. Walsh, "An Introduction to Stochastic Partial Differential Equations," in "Lecture Notes in Mathematics no. 1180", Springer Verlag, 1986, 265-439. doi: 10.1007/BFb0074920. [23] J. B. Walsh, Finite element methods for parabolic stochastic PDE's, Potential Analysis, 23 (2005), 1-43. doi: 10.1007/s11118-004-2950-y.
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