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On the Backward Euler Approximation of the Stochastic Allen-Cahn Equation

Part of: Stochastic analysis Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  30 January 2018

Mihály Kovács ,
Stig Larsson  and
Fredrik Lindgren
Mihály Kovács*
Affiliation:
University of Otago
Stig Larsson*
Affiliation:
Chalmers University of Technology and University of Gothenburg
Fredrik Lindgren*
Affiliation:
Chalmers University of Technology and University of Gothenburg
*
∗ Postal address: Department of Mathematics and Statistics, University of Otago, PO Box 56, Dunedin, New Zealand. Email address: mkovacs@maths.otago.ac.nz
∗∗ Postal address: Department of Mathematical Sciences, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden.
∗∗ Postal address: Department of Mathematical Sciences, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden.
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Abstract

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We consider the stochastic Allen-Cahn equation perturbed by smooth additive Gaussian noise in a spatial domain with smooth boundary in dimension d ≤ 3, and study the semidiscretization in time of the equation by an implicit Euler method. We show that the method converges pathwise with a rate O(Δtγ) for any γ < ½. We also prove that the scheme converges uniformly in the strong Lp-sense but with no rate given.

Keywords

Stochastic partial differential equationAllen-Cahn equationadditive noiseWiener processEuler methodpathwise convergencestrong convergencefactorization method

MSC classification

Primary: 60H15: Stochastic partial differential equations
Secondary: 60H35: Computational methods for stochastic equations 65C30: Stochastic differential and integral equations
Type
Research Article
Information
Journal of Applied Probability , Volume 52 , Issue 2 , June 2015 , pp. 323 - 338
Copyright
© Applied Probability Trust 

References

Alabert, A. and Gyöngy, I. (2006). On numerical approximation of stochastic Burgers' equation. In From Stochastic Calculus to Mathematical Finance. Springer, Berlin, pp. 1–15.Google Scholar
Blömker, D. and Jentzen, A. (2013). Galerkin approximations for the stochastic Burgers equation. SIAM J. Numer. Anal. 51, 694–715.CrossRefGoogle Scholar
Blömker, D., Kamrani, M. and Hosseini, S. M. (2013). Full discretization of the stochastic Burgers equation with correlated noise. IMA J. Numer. Anal. 33, 825–848.Google Scholar
Cioica, P. et al. (2012). On the convergence analysis of Rothe's method. Preprint 124, DFG-Schwerpunktprogramm 1324.Google Scholar
Cox, S. and van Neerven, J. (2013). Pathwise Hölder convergence of the implicit-linear Euler scheme for semi-linear SPDEs with multiplicative noise. Numer. Math. 125, 259–345.CrossRefGoogle Scholar
Da Prato, G. and Debussche, A. (1996). Stochastic Cahn–Hilliard equation. Nonlinear Anal. 26, 241–263.Google Scholar
Da Prato, G. and Zabczyk, J. (1992). Stochastic Equations in Infinite Dimensions (Encyclopedia Math. Appl. 44). Cambridge University Press.CrossRefGoogle Scholar
Elliott, C. M. and Larsson, S. (1992). Error estimates with smooth and nonsmooth data for a finite element method for the Cahn–Hilliard equation. Math. Comp. 58, 603–630, S33–S36.Google Scholar
Gyöngy, I. and Millet, A. (2005). On discretization schemes for stochastic evolution equations. Potential Anal. 23, 99–134.Google Scholar
Gyöngy, I. and Millet, A. (2007). Rate of convergence of implicit approximations for stochastic evolution equations. In Stochastic Differential Equations: Theory and Applications. (Interdiscip. Math. Sci. 2), World Scientific, Hackensack, NJ, pp. 281–310.CrossRefGoogle Scholar
Hausenblas, E. (2002). Numerical analysis of semilinear stochastic evolution equations in Banach spaces. J. Comput. Appl. Math. 147, 485–516.Google Scholar
Hausenblas, E. (2003). Approximation for semilinear stochastic evolution equations. Potential Anal. 18, 141–186.CrossRefGoogle Scholar
Higham, D. J., Mao, X. and Stuart, A. M. (2002). Strong convergence of Euler-type methods for nonlinear stochastic differential equations. SIAM J. Numer. Anal. 40, 1041–1063.Google Scholar
Jentzen, A. (2009). Pathwise numerical approximation of SPDEs with additive noise under non-global Lipschitz coefficients. Potential Anal. 31, 375–404.Google Scholar
Kallenberg, O. (1997). Foundations of Modern Probability. Springer, New York.Google Scholar
Kovács, M., Larsson, S. and Mesforush, A. (2011). Finite element approximation of the Cahn–Hilliard–Cook equation. SIAM J. Numer. Anal. 49, 2407–2429.Google Scholar
Kovács, M., Larsson, S. and Urban, K. (2013). On wavelet–Galerkin methods for semilinear parabolic equations with additive noise. In Monte Carlo and Quasi-Monte Carlo Methods 2012, Springer, Heidelberg, pp. 481–499.CrossRefGoogle Scholar
Kunita, H. (1990). Stochastic Flows and Stochastic Differential Equations (Camb. Stud. Adv. Math. 24). Cambridge University Press.Google Scholar
Liu, D. (2003). Convergence of the spectral method for stochastic Ginzburg–Landau equation driven by space–time white noise. Commun. Math. Sci. 1, 361–375.Google Scholar
Liu, W. (2010). Invariance of subspaces under the solution flow of SPDE. Infin. Dimens. Anal. Quantum Prob. Relat. Top. 13, 87–98.CrossRefGoogle Scholar
Liu, W. (2013). Well-posedness of stochastic partial differential equations with Lyapunov condition. J. Differential Equat. 255, 572–592.Google Scholar
Liu, W. and Röckner, M. (2010). S{PDE} in Hilbert space with locally monotone coefficients. J. Funct. Anal. 259, 2902–2922.CrossRefGoogle Scholar
Pazy, A. (1983). Semigroups of Linear Operators and Applications to Partial Differential Equations (Appl. Math. Sci. 44). Springer, New York.Google Scholar
Prévôt, C. and Röckner, M. (2007). A Concise Course on Stochastic Partial Differential Equations (Lecture Notes Math. 1905). Springer, Berlin.Google Scholar
Printems, J. (2001). On the discretization in time of parabolic stochastic partial differential equations. M2AN Math. Model. Numer. Anal. 35, 1055–1078.Google Scholar
Sauer, M. and Stannat, W. (2015). Lattice approximation for stochastic reaction diffusion equations with one-sided Lipschitz condition. Math. Comp. 84, 743–766.Google Scholar
Thomée, V. (2006). Galerkin Finite Element Methods for Parabolic Problems (Springer Ser. Comput. Math. 25), 2nd edn. Springer, Berlin.Google Scholar