# Fully-discrete finite element approximations for a fourth-order linear stochastic parabolic equation with additive space-time white noise

Georgios T. Kossioris; Georgios E. Zouraris

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

- Volume: 44, Issue: 2, page 289-322
- ISSN: 0764-583X

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topKossioris, Georgios T., and 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 44.2 (2010): 289-322. <http://eudml.org/doc/250856>.

@article{Kossioris2010,

abstract = {
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.
},

author = {Kossioris, Georgios T., Zouraris, Georgios E.},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Finite element method; space-time white noise; Backward
Euler time-stepping; fully-discrete approximations; a priori error
estimates; finite element method; backward Euler time-stepping; initial-boundary value problem; fourth-order linear stochastic parabolic equation},

language = {eng},

month = {3},

number = {2},

pages = {289-322},

publisher = {EDP Sciences},

title = {Fully-discrete finite element approximations for a fourth-order linear stochastic parabolic equation with additive space-time white noise},

url = {http://eudml.org/doc/250856},

volume = {44},

year = {2010},

}

TY - JOUR

AU - Kossioris, Georgios T.

AU - Zouraris, Georgios E.

TI - Fully-discrete finite element approximations for a fourth-order linear stochastic parabolic equation with additive space-time white noise

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2010/3//

PB - EDP Sciences

VL - 44

IS - 2

SP - 289

EP - 322

AB -
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.

LA - eng

KW - Finite element method; space-time white noise; Backward
Euler time-stepping; fully-discrete approximations; a priori error
estimates; finite element method; backward Euler time-stepping; initial-boundary value problem; fourth-order linear stochastic parabolic equation

UR - http://eudml.org/doc/250856

ER -

## References

top- 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. Zbl0907.65147
- 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. Zbl1080.65003
- L. Bin, Numerical method for a parabolic stochastic partial differential equation. Master Thesis 2004-03, Chalmers University of Technology, Göteborg, Sweden (2004).
- 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. Zbl0201.07803
- S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. Springer-Verlag, New York, USA (1994). Zbl0804.65101
- 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).
- C. Cardon-Weber, Cahn-Hilliard equation: existence of the solution and of its density. Bernoulli7 (2001) 777–816. Zbl0995.60058
- P.G. Ciarlet, The finite element methods for elliptic problems. North-Holland, New York (1987).
- H. Cook, Browian motion in spinodal decomposition. Acta Metall.18 (1970) 297–306.
- G. Da Prato and A. Debussche, Stochastic Cahn-Hilliard equation. Nonlinear Anal.26 (1996) 241–263. Zbl0838.60056
- 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). Zbl0635.47002
- K.R. Elder, T.M. Rogers and R.C. Desai, Numerical study of the late stages of spinodal decomposition. Phys. Rev. B37 (1987) 9638–9649.
- G.H. Golub and C.F. Van Loan, Matrix Computations. Second Edition, The John Hopkins University Press, Baltimore, USA (1989). Zbl0733.65016
- W. Grecksch and P.E. Kloeden, Time-discretised Galerkin approximations of parabolic stochastic PDEs. Bull. Austral. Math. Soc.54 (1996) 79–85. Zbl0880.35143
- E. Hausenblas, Numerical analysis of semilinear stochastic evolution equations in Banach spaces. J. Comput. Appl. Math.147 (2002) 485–516. Zbl1026.65005
- E. Hausenblas, Approximation for semilinear stochastic evolution equations. Potential Anal.18 (2003) 141–186. Zbl1015.60053
- G. Kallianpur and J. Xiong, Stochastic Differential Equations in Infinite Dimensional Spaces, Lecture Notes-Monograph Series26. Institute of Mathematical Statistics, Hayward, USA (1995).
- L. Kielhorn and M. Muthukumar, Spinodal decomposition of symmetric diblock copolymer homopolymer blends at the Lifshitz point. J. Chem. Phys.110 (1999) 4079–4089.
- 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. Zbl0988.60066
- 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). Zbl1189.65018
- J.L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol. I. Springer-Verlag, Berlin-Heidelberg, Germany (1972). Zbl0223.35039
- 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. Zbl1136.60044
- J. Printems, On the discretization in time of parabolic stochastic partial differential equations. ESAIM: M2AN35 (2001) 1055–1078. Zbl0991.60051
- V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Spriger Series in Computational Mathematics25. Springer-Verlag, Berlin-Heidelberg, Germany (1997). Zbl0884.65097
- 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).
- Y. Yan, Semidiscrete Galerkin approximation for a linear stochastic parabolic partial differential equation driven by an additive noise. BIT44 (2004) 829–847. Zbl1080.65006
- Y. Yan, Galerkin finite element methods for stochastic parabolic partial differential equations. SIAM J. Numer. Anal.43 (2005) 1363–1384. Zbl1112.60049
- J.B. Walsh, An introduction to stochastic partial differential equations., Lecture Notes in Mathematics1180. Springer Verlag, Berlin-Heidelberg, Germany (1986) 265–439.
- J.B. Walsh, Finite element methods for parabolic stochastic PDEs. Potential Anal.23 (2005) 1–43. Zbl1065.60082

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