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.
@article{M2AN_2010__44_2_289_0,
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},
volume = {44},
year = {2010},
pages = {289-322},
doi = {10.1051/m2an/2010003},
mrnumber = {2655951},
zbl = {1189.65018},
language = {en},
url = {http://dml.mathdoc.fr/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, Tome 44 (2010) pp. 289-322. doi : 10.1051/m2an/2010003. http://gdmltest.u-ga.fr/item/M2AN_2010__44_2_289_0/
[1] , and , Finite element and difference approximation of some linear stochastic partial differential equations. Stoch. Stoch. Rep. 64 (1998) 117-142. | Zbl 0907.65147
[2] , and , Galerkin finite element approximations of stochastic elliptic partial differential equations. SIAM J. Numer. Anal. 42 (2004) 800-825. | Zbl 1080.65003
[3] , Numerical method for a parabolic stochastic partial differential equation. Master Thesis 2004-03, Chalmers University of Technology, Göteborg, Sweden (2004).
[4] and , 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] and , The Mathematical Theory of Finite Element Methods. Springer-Verlag, New York, USA (1994). | Zbl 1135.65042
[6] , 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] , Cahn-Hilliard equation: existence of the solution and of its density. Bernoulli 7 (2001) 777-816. | Zbl 0995.60058
[8] , The finite element methods for elliptic problems. North-Holland, New York (1987). | Zbl 0383.65058
[9] , Browian motion in spinodal decomposition. Acta Metall. 18 (1970) 297-306.
[10] and , Stochastic Cahn-Hilliard equation. Nonlinear Anal. 26 (1996) 241-263. | Zbl 0838.60056
[11] and , 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] , and , Numerical study of the late stages of spinodal decomposition. Phys. Rev. B 37 (1987) 9638-9649.
[13] and , Matrix Computations. Second Edition, The John Hopkins University Press, Baltimore, USA (1989). | Zbl 0592.65011
[14] and , Time-discretised Galerkin approximations of parabolic stochastic PDEs. Bull. Austral. Math. Soc. 54 (1996) 79-85. | Zbl 0880.35143
[15] , Numerical analysis of semilinear stochastic evolution equations in Banach spaces. J. Comput. Appl. Math. 147 (2002) 485-516. | Zbl 1026.65005
[16] , Approximation for semilinear stochastic evolution equations. Potential Anal. 18 (2003) 141-186. | Zbl 1015.60053
[17] and , Stochastic Differential Equations in Infinite Dimensional Spaces, Lecture Notes-Monograph Series 26. Institute of Mathematical Statistics, Hayward, USA (1995). | Zbl 0859.60050
[18] and , Spinodal decomposition of symmetric diblock copolymer homopolymer blends at the Lifshitz point. J. Chem. Phys. 110 (1999) 4079-4089.
[19] and , Linear-implicit strong schemes for Itô-Galerkin approximations of stochastic PDEs. J. Appl. Math. Stoch. Anal. 14 (2001) 47-53. | Zbl 0988.60066
[20] and , 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] and , Non-Homogeneous Boundary Value Problems and Applications, Vol. I. Springer-Verlag, Berlin-Heidelberg, Germany (1972). | Zbl 0223.35039
[22] and , Lower bounds and non-uniform time discretization for approximation of stochastic heat equations. Found. Comput. Math. 7 (2007) 135-181. | Zbl 1136.60044
[23] , On the discretization in time of parabolic stochastic partial differential equations. ESAIM: M2AN 35 (2001) 1055-1078. | Numdam | Zbl 0991.60051
[24] , Galerkin Finite Element Methods for Parabolic Problems, Spriger Series in Computational Mathematics 25. Springer-Verlag, Berlin-Heidelberg, Germany (1997). | Zbl 0528.65052
[25] , 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] , 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] , Galerkin finite element methods for stochastic parabolic partial differential equations. SIAM J. Numer. Anal. 43 (2005) 1363-1384. | Zbl 1112.60049
[28] , An introduction to stochastic partial differential equations., Lecture Notes in Mathematics 1180. Springer Verlag, Berlin-Heidelberg, Germany (1986) 265-439. | Zbl 0608.60060
[29] , Finite element methods for parabolic stochastic PDEs. Potential Anal. 23 (2005) 1-43. | Zbl 1065.60082