Fully-discrete finite element approximations for a fourth-order linear stochastic parabolic equation with additive space-time white noise
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 44 (2010) no. 2, p. 289-322

We consider an initial and Dirichlet boundary value problem for a fourth-order linear stochastic parabolic equation, in one space dimension, forced by an additive space-time white noise. Discretizing the space-time white noise a modelling error is introduced and a regularized fourth-order linear stochastic parabolic problem is obtained. Fully-discrete approximations to the solution of the regularized problem are constructed by using, for discretization in space, a Galerkin finite element method based on C0 or C1 piecewise polynomials, and, for time-stepping, the Backward Euler method. We derive strong a priori estimates for the modelling error and for the approximation error to the solution of the regularized problem.

DOI : https://doi.org/10.1051/m2an/2010003
Classification:  65M60,  65M15,  65C20
Keywords: finite element method, space-time white noise, backward Euler time-stepping, fully-discrete approximations, a priori error estimates
     author = {Kossioris, Georgios T. and Zouraris, Georgios E.},
     title = {Fully-discrete finite element approximations for a fourth-order linear stochastic parabolic equation with additive space-time white noise},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {44},
     number = {2},
     year = {2010},
     pages = {289-322},
     doi = {10.1051/m2an/2010003},
     zbl = {1189.65018},
     mrnumber = {2655951},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2010__44_2_289_0}
Kossioris, Georgios T.; Zouraris, Georgios E. Fully-discrete finite element approximations for a fourth-order linear stochastic parabolic equation with additive space-time white noise. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 44 (2010) no. 2, pp. 289-322. doi : 10.1051/m2an/2010003. http://www.numdam.org/item/M2AN_2010__44_2_289_0/

[1] E.J. Allen, S.J. Novosel and Z. Zhang, Finite element and difference approximation of some linear stochastic partial differential equations. Stoch. Stoch. Rep. 64 (1998) 117-142. | Zbl 0907.65147

[2] I. Babuška, R. Tempone and G.E. Zouraris, Galerkin finite element approximations of stochastic elliptic partial differential equations. SIAM J. Numer. Anal. 42 (2004) 800-825. | Zbl 1080.65003

[3] L. Bin, Numerical method for a parabolic stochastic partial differential equation. Master Thesis 2004-03, Chalmers University of Technology, Göteborg, Sweden (2004).

[4] 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. | Zbl 0201.07803

[5] S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. Springer-Verlag, New York, USA (1994). | Zbl 1135.65042

[6] C. Cardon-Weber, Implicit approximation scheme for the Cahn-Hilliard stochastic equation. PMA 613, Laboratoire de Probabilités et Modèles Alétoires, CNRS U.M.R. 7599, Universtités Paris VI et VII, Paris, France (2000). | Zbl 0995.60058

[7] C. Cardon-Weber, Cahn-Hilliard equation: existence of the solution and of its density. Bernoulli 7 (2001) 777-816. | Zbl 0995.60058

[8] P.G. Ciarlet, The finite element methods for elliptic problems. North-Holland, New York (1987). | Zbl 0383.65058

[9] H. Cook, Browian motion in spinodal decomposition. Acta Metall. 18 (1970) 297-306.

[10] G. Da Prato and A. Debussche, Stochastic Cahn-Hilliard equation. Nonlinear Anal. 26 (1996) 241-263. | Zbl 0838.60056

[11] 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, USA (1988). | Zbl 0635.47002

[12] K.R. Elder, T.M. Rogers and R.C. Desai, Numerical study of the late stages of spinodal decomposition. Phys. Rev. B 37 (1987) 9638-9649.

[13] G.H. Golub and C.F. Van Loan, Matrix Computations. Second Edition, The John Hopkins University Press, Baltimore, USA (1989). | Zbl 0592.65011

[14] W. Grecksch and P.E. Kloeden, Time-discretised Galerkin approximations of parabolic stochastic PDEs. Bull. Austral. Math. Soc. 54 (1996) 79-85. | Zbl 0880.35143

[15] E. Hausenblas, Numerical analysis of semilinear stochastic evolution equations in Banach spaces. J. Comput. Appl. Math. 147 (2002) 485-516. | Zbl 1026.65005

[16] E. Hausenblas, Approximation for semilinear stochastic evolution equations. Potential Anal. 18 (2003) 141-186. | Zbl 1015.60053

[17] G. Kallianpur and J. Xiong, Stochastic Differential Equations in Infinite Dimensional Spaces, Lecture Notes-Monograph Series 26. Institute of Mathematical Statistics, Hayward, USA (1995). | Zbl 0859.60050

[18] L. Kielhorn and M. Muthukumar, Spinodal decomposition of symmetric diblock copolymer homopolymer blends at the Lifshitz point. J. Chem. Phys. 110 (1999) 4079-4089.

[19] P.E. Kloeden and S. Shot, Linear-implicit strong schemes for Itô-Galerkin approximations of stochastic PDEs. J. Appl. Math. Stoch. Anal. 14 (2001) 47-53. | Zbl 0988.60066

[20] 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. TRITA-NA 2008:2, School of Computer Science and Communication, KTH, Stockholm, Sweden (2008). | Zbl 1189.65018

[21] J.L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol. I. Springer-Verlag, Berlin-Heidelberg, Germany (1972). | Zbl 0223.35039

[22] T. Müller-Gronbach and K. Ritter, Lower bounds and non-uniform time discretization for approximation of stochastic heat equations. Found. Comput. Math. 7 (2007) 135-181. | Zbl 1136.60044

[23] J. Printems, On the discretization in time of parabolic stochastic partial differential equations. ESAIM: M2AN 35 (2001) 1055-1078. | Numdam | Zbl 0991.60051

[24] V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Spriger Series in Computational Mathematics 25. Springer-Verlag, Berlin-Heidelberg, Germany (1997). | Zbl 0528.65052

[25] Y. Yan, Error analysis and smothing properies of discretized deterministic and stochastic parabolic problems. Ph.D. Thesis, Department of Computational Mathematics, Chalmers University of Technology and Göteborg University, Göteborg, Sweden (2003).

[26] Y. Yan, Semidiscrete Galerkin approximation for a linear stochastic parabolic partial differential equation driven by an additive noise. BIT 44 (2004) 829-847. | Zbl 1080.65006

[27] Y. Yan, Galerkin finite element methods for stochastic parabolic partial differential equations. SIAM J. Numer. Anal. 43 (2005) 1363-1384. | Zbl 1112.60049

[28] J.B. Walsh, An introduction to stochastic partial differential equations., Lecture Notes in Mathematics 1180. Springer Verlag, Berlin-Heidelberg, Germany (1986) 265-439. | Zbl 0608.60060

[29] J.B. Walsh, Finite element methods for parabolic stochastic PDEs. Potential Anal. 23 (2005) 1-43. | Zbl 1065.60082