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 & 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.  doi: 10.1080/17442509808834159.  Google Scholar

[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.  doi: 10.1007/s11401-009-0219-x.  Google Scholar

[3]

L. Bin, "Numerical Method for a Parabolic Stochastic Partial Differential Equation,", Master Thesis 2004-03, (2004), 2004.   Google Scholar

[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.  doi: 10.1090/S0002-9947-07-04387-5.  Google Scholar

[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.  doi: 10.1137/0707006.  Google Scholar

[6]

A. Debussche and L. Zambotti, Conservative stochastic Cahn-Hilliard equation with reflection,, Annals of Probability, 35 (2007), 1706.  doi: 10.1214/009117906000000773.  Google Scholar

[7]

N. Dunford and J. T. Schwartz, "Linear Operators. Part II. Spectral Theory. Self Adjoint Operators in Hilbert Space,", Reprint of the 1963 original, (1963).   Google Scholar

[8]

W. Grecksch and P. E. Kloeden, Time-discretised Galerkin approximations of parabolic stochastic PDE's,, Bull. Austral. Math. Soc., 54 (1996), 79.  doi: 10.1017/S0004972700015094.  Google Scholar

[9]

G. H. Golub and C. F. Van Loan, "Matrix Computations,", Second Edition, (1989).   Google Scholar

[10]

P. C. Hohenberg and B. I. Halperin, Theory of dynamic critical phenomena,, J. Rev. Mod. Phys., 49 (1977), 435.  doi: 10.1103/RevModPhys.49.435.  Google Scholar

[11]

G. Kallianpur and J. Xiong, "Stochastic Differential Equations in Infinite Dimensional Spaces,", Institute of Mathematical Statistics, 26 (1995).   Google Scholar

[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.  doi: 10.1063/1.1616513.  Google Scholar

[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.  doi: 10.1155/S1048953301000053.  Google Scholar

[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.  doi: 10.1051/m2an/2010003.  Google Scholar

[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.  doi: 10.1016/j.apnum.2012.01.003.  Google Scholar

[16]

S. Larsson and A. Mesforush, Finite element approximation of the linearized Cahn-Hilliard-Cook equation,, IMA J. Numer. Anal., 31 (2011), 1315.  doi: 10.1093/imanum/drq042.  Google Scholar

[17]

J. L. Lions and E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications. Vol. I,", Springer-Verlag, (1972).   Google Scholar

[18]

J. Printems, On the discretization in time of parabolic stochastic partial differential equations,, Mathematical Modelling and Numerical Analysis, 35 (2001), 1055.  doi: 10.1051/m2an:2001148.  Google Scholar

[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.  doi: 10.1103/PhysRevB.37.9638.  Google Scholar

[20]

V. Thomée, "Galerkin Finite Element Methods for Parabolic Problems,", Spriger Series in Computational Mathematics, (1997).   Google Scholar

[21]

Y. Yan, Galerkin finite element methods for stochastic parabolic partial differential equations,, SIAM Journal on Numerical Analysis, 43 (2005), 1363.  doi: 10.1137/040605278.  Google Scholar

[22]

J. B. Walsh, "An Introduction to Stochastic Partial Differential Equations,", in, (1180), 265.  doi: 10.1007/BFb0074920.  Google Scholar

[23]

J. B. Walsh, Finite element methods for parabolic stochastic PDE's,, Potential Analysis, 23 (2005), 1.  doi: 10.1007/s11118-004-2950-y.  Google Scholar

show all references

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.  doi: 10.1080/17442509808834159.  Google Scholar

[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.  doi: 10.1007/s11401-009-0219-x.  Google Scholar

[3]

L. Bin, "Numerical Method for a Parabolic Stochastic Partial Differential Equation,", Master Thesis 2004-03, (2004), 2004.   Google Scholar

[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.  doi: 10.1090/S0002-9947-07-04387-5.  Google Scholar

[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.  doi: 10.1137/0707006.  Google Scholar

[6]

A. Debussche and L. Zambotti, Conservative stochastic Cahn-Hilliard equation with reflection,, Annals of Probability, 35 (2007), 1706.  doi: 10.1214/009117906000000773.  Google Scholar

[7]

N. Dunford and J. T. Schwartz, "Linear Operators. Part II. Spectral Theory. Self Adjoint Operators in Hilbert Space,", Reprint of the 1963 original, (1963).   Google Scholar

[8]

W. Grecksch and P. E. Kloeden, Time-discretised Galerkin approximations of parabolic stochastic PDE's,, Bull. Austral. Math. Soc., 54 (1996), 79.  doi: 10.1017/S0004972700015094.  Google Scholar

[9]

G. H. Golub and C. F. Van Loan, "Matrix Computations,", Second Edition, (1989).   Google Scholar

[10]

P. C. Hohenberg and B. I. Halperin, Theory of dynamic critical phenomena,, J. Rev. Mod. Phys., 49 (1977), 435.  doi: 10.1103/RevModPhys.49.435.  Google Scholar

[11]

G. Kallianpur and J. Xiong, "Stochastic Differential Equations in Infinite Dimensional Spaces,", Institute of Mathematical Statistics, 26 (1995).   Google Scholar

[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.  doi: 10.1063/1.1616513.  Google Scholar

[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.  doi: 10.1155/S1048953301000053.  Google Scholar

[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.  doi: 10.1051/m2an/2010003.  Google Scholar

[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.  doi: 10.1016/j.apnum.2012.01.003.  Google Scholar

[16]

S. Larsson and A. Mesforush, Finite element approximation of the linearized Cahn-Hilliard-Cook equation,, IMA J. Numer. Anal., 31 (2011), 1315.  doi: 10.1093/imanum/drq042.  Google Scholar

[17]

J. L. Lions and E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications. Vol. I,", Springer-Verlag, (1972).   Google Scholar

[18]

J. Printems, On the discretization in time of parabolic stochastic partial differential equations,, Mathematical Modelling and Numerical Analysis, 35 (2001), 1055.  doi: 10.1051/m2an:2001148.  Google Scholar

[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.  doi: 10.1103/PhysRevB.37.9638.  Google Scholar

[20]

V. Thomée, "Galerkin Finite Element Methods for Parabolic Problems,", Spriger Series in Computational Mathematics, (1997).   Google Scholar

[21]

Y. Yan, Galerkin finite element methods for stochastic parabolic partial differential equations,, SIAM Journal on Numerical Analysis, 43 (2005), 1363.  doi: 10.1137/040605278.  Google Scholar

[22]

J. B. Walsh, "An Introduction to Stochastic Partial Differential Equations,", in, (1180), 265.  doi: 10.1007/BFb0074920.  Google Scholar

[23]

J. B. Walsh, Finite element methods for parabolic stochastic PDE's,, Potential Analysis, 23 (2005), 1.  doi: 10.1007/s11118-004-2950-y.  Google Scholar

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