Weak convergence for a spatial approximation of the nonlinear stochastic heat equation

Andersson, Adam; Larsson, Stig

doi:10.1090/mcom/3016

Weak convergence for a spatial approximation of the nonlinear stochastic heat equation
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by Adam Andersson and Stig Larsson PDF

Math. Comp. 85 (2016), 1335-1358 Request permission

Abstract:

We find the weak rate of convergence of the spatially semidiscrete finite element approximation of the nonlinear stochastic heat equation. Both multiplicative and additive noise is considered under different assumptions. This extends an earlier result of Debussche in which time discretization is considered for the stochastic heat equation perturbed by white noise. It is known that this equation has a solution only in one space dimension. In order to obtain results for higher dimensions, colored noise is considered here, besides white noise in one dimension. Integration by parts in the Malliavin sense is used in the proof. The rate of weak convergence is, as expected, essentially twice the rate of strong convergence.

References

Charles-Edouard Bréhier, Approximation of the invariant measure with an Euler scheme for stochastic PDEs driven by space-time white noise, Potential Anal. 40 (2014), no. 1, 1–40. MR 3146507, DOI 10.1007/s11118-013-9338-9
Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, 3rd ed., Texts in Applied Mathematics, vol. 15, Springer, New York, 2008. MR 2373954, DOI 10.1007/978-0-387-75934-0
Giuseppe Da Prato and Jerzy Zabczyk, Stochastic equations in infinite dimensions, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1992. MR 1207136, DOI 10.1017/CBO9780511666223
Anne de Bouard and Arnaud Debussche, Weak and strong order of convergence of a semidiscrete scheme for the stochastic nonlinear Schrödinger equation, Appl. Math. Optim. 54 (2006), no. 3, 369–399. MR 2268663, DOI 10.1007/s00245-006-0875-0
Arnaud Debussche, Weak approximation of stochastic partial differential equations: the nonlinear case, Math. Comp. 80 (2011), no. 273, 89–117. MR 2728973, DOI 10.1090/S0025-5718-2010-02395-6
Arnaud Debussche and Jacques Printems, Weak order for the discretization of the stochastic heat equation, Math. Comp. 78 (2009), no. 266, 845–863. MR 2476562, DOI 10.1090/S0025-5718-08-02184-4
Nelson Dunford and Jacob T. Schwartz, Linear operators. Part II, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. Spectral theory. Selfadjoint operators in Hilbert space; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1963 original; A Wiley-Interscience Publication. MR 1009163
A. Grorud and É. Pardoux, Intégrales hilbertiennes anticipantes par rapport à un processus de Wiener cylindrique et calcul stochastique associé, Appl. Math. Optim. 25 (1992), no. 1, 31–49 (French, with English summary). MR 1133251, DOI 10.1007/BF01184155
Erika Hausenblas, Weak approximation for semilinear stochastic evolution equations, Stochastic analysis and related topics VIII, Progr. Probab., vol. 53, Birkhäuser, Basel, 2003, pp. 111–128. MR 2189620
Erika Hausenblas, Weak approximation of the stochastic wave equation, J. Comput. Appl. Math. 235 (2010), no. 1, 33–58. MR 2671031, DOI 10.1016/j.cam.2010.03.026
Svante Janson, Gaussian Hilbert spaces, Cambridge Tracts in Mathematics, vol. 129, Cambridge University Press, Cambridge, 1997. MR 1474726, DOI 10.1017/CBO9780511526169
Arnulf Jentzen and Michael Röckner, Regularity analysis for stochastic partial differential equations with nonlinear multiplicative trace class noise, J. Differential Equations 252 (2012), no. 1, 114–136. MR 2852200, DOI 10.1016/j.jde.2011.08.050
Mihály Kovács, Stig Larsson, and Fredrik Lindgren, Weak convergence of finite element approximations of linear stochastic evolution equations with additive noise, BIT 52 (2012), no. 1, 85–108. MR 2891655, DOI 10.1007/s10543-011-0344-2
Mihály Kovács, Stig Larsson, and Fredrik Lindgren, Weak convergence of finite element approximations of linear stochastic evolution equations with additive noise II. Fully discrete schemes, BIT 53 (2013), no. 2, 497–525. MR 3123856, DOI 10.1007/s10543-012-0405-1
R. Kruse, Strong and Weak Approximation of Semilinear Stochastic Evolution Equations, PhD thesis, University of Bielefeld, 2012.
Raphael Kruse, Optimal error estimates of Galerkin finite element methods for stochastic partial differential equations with multiplicative noise, IMA J. Numer. Anal. 34 (2014), no. 1, 217–251. MR 3168284, DOI 10.1093/imanum/drs055
Raphael Kruse and Stig Larsson, Optimal regularity for semilinear stochastic partial differential equations with multiplicative noise, Electron. J. Probab. 17 (2012), no. 65, 19. MR 2968672, DOI 10.1214/EJP.v17-2240
Jorge A. León and David Nualart, Stochastic evolution equations with random generators, Ann. Probab. 26 (1998), no. 1, 149–186. MR 1617045, DOI 10.1214/aop/1022855415
Felix Lindner and René L. Schilling, Weak order for the discretization of the stochastic heat equation driven by impulsive noise, Potential Anal. 38 (2013), no. 2, 345–379. MR 3015355, DOI 10.1007/s11118-012-9276-y
Alessandra Lunardi, Interpolation theory, 2nd ed., Appunti. Scuola Normale Superiore di Pisa (Nuova Serie). [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)], Edizioni della Normale, Pisa, 2009. MR 2523200
David Nualart, The Malliavin calculus and related topics, 2nd ed., Probability and its Applications (New York), Springer-Verlag, Berlin, 2006. MR 2200233
Vidar Thomée, Galerkin finite element methods for parabolic problems, 2nd ed., Springer Series in Computational Mathematics, vol. 25, Springer-Verlag, Berlin, 2006. MR 2249024
J. M. A. M. van Neerven, Stochastic Evolution Equations, ISEM lecture notes (2008).
Xiaojie Wang and Siqing Gan, Weak convergence analysis of the linear implicit Euler method for semilinear stochastic partial differential equations with additive noise, J. Math. Anal. Appl. 398 (2013), no. 1, 151–169. MR 2984323, DOI 10.1016/j.jmaa.2012.08.038