×

Strong convergence of a fully discrete finite element approximation of the stochastic Cahn-Hilliard equation. (English) Zbl 1390.60231

Summary: We consider the stochastic Cahn-Hilliard equation driven by additive Gaussian noise in a convex domain with polygonal boundary in dimension \(d\leq 3\). We discretize the equation using a standard finite element method in space and a fully implicit backward Euler method in time. By proving optimal error estimates on subsets of the probability space with arbitrarily large probability and uniform-in-time moment bounds we show that the numerical solution converges strongly to the solution as the discretization parameters tend to zero.

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
65C30 Numerical solutions to stochastic differential and integral equations
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] S. Becker and A. Jentzen, {\it Strong Convergence Rates for Nonlinearity-Truncated Euler-Type Approximations of Stochastic Ginzburg-Landau Equations}, preprint, , 2016. · Zbl 1403.60052
[2] D. Blömker and A. Jentzen, {\it Galerkin approximations for the stochastic Burgers equation}, SIAM J. Numer. Anal., 51 (2013), pp. 694-715, . · Zbl 1267.60071
[3] D. Blömker, M. Kamrani, and S. M. Hosseini, {\it Full discretization of the stochastic Burgers equation with correlated noise}, IMA J. Numer. Anal., 33 (2013), pp. 825-848, . · Zbl 1280.65008
[4] D. Blömker, S. Maier-Paape, and T. Wanner, {\it Spinodal decomposition for the stochastic Cahn-Hilliard equation}, in International Conference on Differential Equations, Vols. 1, 2 (Berlin, 1999), World Scientific Publishing, River Edge, NJ, 2000, pp. 1265-1267. · Zbl 0966.60057
[5] Z. Brzeźniak, E. Carelli, and A. Prohl, {\it Finite-element-based discretizations of the incompressible Navier-Stokes equations with multiplicative random forcing}, IMA J. Numer. Anal., 33 (2013), pp. 771-824, . · Zbl 1426.76227
[6] C. Cardon-Weber, {\it Implicit Approximation Scheme for the Cahn-Hilliard Stochastic Equation}. preprint, Laboratoire des Probabilités et Modelèles Aléatoires, Université Paris V, 2000.
[7] E. Carelli and A. Prohl, {\it Rates of convergence for discretizations of the stochastic incompressible Navier-Stokes equations}, SIAM J. Numer. Anal., 50 (2012), pp. 2467-2496, . · Zbl 1426.76231
[8] H. E. Cook, {\it Brownian motion in spinodal decomposition}, Acta Metallurgica, 18 (1970).
[9] G. Da Prato and A. Debussche, {\it Stochastic Cahn-Hilliard equation}, Nonlinear Anal., 26 (1996), pp. 241-263, . · Zbl 0838.60056
[10] G. Da Prato and J. Zabczyk, {\it Stochastic Equations in Infinite Dimensions}, Encyclopedia of Mathematics and Its Applications, 44, Cambridge University Press, Cambridge, 1992. · Zbl 0761.60052
[11] \sc E. B. Davies, {\it Spectral Theory and Differential Operators}, Cambridge Studies in Advanced Mathematics 42, Cambridge University Press, Cambridge, 1995. · Zbl 0893.47004
[12] C. M. Elliott and S. Larsson, {\it Error estimates with smooth and nonsmooth data for a finite element method for the Cahn-Hilliard equation}, Math. Comp., 58 (1992), pp. 603-630, S33-S36, . · Zbl 0762.65075
[13] G. B. Folland, {\it Real Analysis}, John Wiley & Sons, New York, 1999. · Zbl 0924.28001
[14] I. Gyöngy and A. Millet, {\it On discretization schemes for stochastic evolution equations}, Potential Anal., 23 (2005), pp. 99-134, . · Zbl 1067.60049
[15] I. Gyöngy, S. Sabanis, and D. Šiška, {\it Convergence of tamed Euler schemes for a class of stochastic evolution equations}, Stoch. Partial Differ. Equ. Anal. Comput., 4 (2016), pp. 225-245, . · Zbl 1342.60115
[16] M. Hutzenthaler and A. Jentzen, {\it On a Perturbation Theory and on Strong Convergence Rates for Stochastic Ordinary and Partial Differential Equations with Non-Globally Monotone Coefficients}, preprint, , 2014. · Zbl 07206753
[17] M. Hutzenthaler, A. Jentzen, and D. Salimova, {\it Strong Convergence of Full-Discrete Nonlinearity-Truncated Accelerated Exponential Euler-Type Approximations for Stochastic Kuramoto-Sivashinsky Equations}, preprint, , 2016. · Zbl 1406.60102
[18] A. Jentzen and P. Pu\vsnik, {\it Strong Convergence Rates for an Explicit Numerical Approximation Method for Stochastic Evolution Equations with Non-Globally Lipschitz Continuous Nonlinearities}, preprint, , 2015. · Zbl 1330.60084
[19] G. T. Kossioris and G. E. Zouraris, {\it Finite element approximations for a linear fourth-order parabolic SPDE in two and three space dimensions with additive space-time white noise}, Appl. Numer. Math., 67 (2013), pp. 243-261, . · Zbl 1271.65016
[20] M. Kovács, S. Larsson, and F. Lindgren, {\it On the backward Euler approximation of the stochastic Allen-Cahn equation.}, J. Appl. Probab., 52 (2015), pp. 323-338, . · Zbl 1323.60089
[21] M. Kovács, S. Larsson, and F. Lindgren, {\it On the discretisation in time of the stochastic Allen-Cahn equation}, Math. Nachr., (2018), pp. 1-30, . · Zbl 1388.60113
[22] M. Kovács, S. Larsson, and A. Mesforush, {\it Finite element approximation of the Cahn-Hilliard-Cook equation}, SIAM J. Numer. Anal., 49 (2011), pp. 2407-2429, . · Zbl 1248.65012
[23] M. Kovács, S. Larsson, and A. Mesforush, {\it Erratum: Finite element approximation of the Cahn-Hilliard-Cook equation}, SIAM J. Numer. Anal., 52 (2014), pp. 2594-2597, . · Zbl 1305.65015
[24] R. Kurniawan, {\it Numerical approximations of stochastic partial differential equations with non-globally Lipschitz continuous nonlinearities}, Master’s thesis, ETH Zürich, 2014.
[25] S. Larsson and A. Mesforush, {\it Finite-element approximation of the linearized Cahn-Hilliard-Cook equation}, IMA J. Numer. Anal., 31 (2011), pp. 1315-1333, . · Zbl 1236.65126
[26] J. Printems, {\it On the discretization in time of parabolic stochastic partial differential equations}, M2AN Math. Model. Numer. Anal., 35 (2001), pp. 1055-1078, . · Zbl 0991.60051
[27] V. Thomée, {\it Galerkin Finite Element Methods for Parabolic Problems}, Springer-Verlag, Berlin, 2006. · Zbl 1105.65102
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.