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


Authors: Adam Andersson and Stig Larsson
Journal: Math. Comp. 85 (2016), 1335-1358
MSC (2010): Primary 65M60, 60H15, 60H35, 65C30
DOI: https://doi.org/10.1090/mcom/3016
Published electronically: August 20, 2015
MathSciNet review: 3454367
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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.


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  • [1] 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, https://doi.org/10.1007/s11118-013-9338-9
  • [2] 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 (2008m:65001)
  • [3] 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 (95g:60073)
  • [4] 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 (2008g:60208), https://doi.org/10.1007/s00245-006-0875-0
  • [5] Arnaud Debussche, Weak approximation of stochastic partial differential equations: the nonlinear case, Math. Comp. 80 (2011), no. 273, 89-117. MR 2728973 (2011j:65014), https://doi.org/10.1090/S0025-5718-2010-02395-6
  • [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 (2010f:60192), https://doi.org/10.1090/S0025-5718-08-02184-4
  • [7] N. Dunford and J. T. Schwartz, Linear Operators. Part II, Wiley Classics Library, John Wiley & Sons Inc., New York, 1988. Reprint of 1963 original. MR 1009163 (90g:47001b)
  • [8] 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 (93b:60114), https://doi.org/10.1007/BF01184155
  • [9] 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 (2006k:60114)
  • [10] Erika Hausenblas, Weak approximation of the stochastic wave equation, J. Comput. Appl. Math. 235 (2010), no. 1, 33-58. MR 2671031 (2011i:60109), https://doi.org/10.1016/j.cam.2010.03.026
  • [11] Svante Janson, Gaussian Hilbert Spaces, Cambridge Tracts in Mathematics, vol. 129, Cambridge University Press, Cambridge, 1997. MR 1474726 (99f:60082)
  • [12] 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, https://doi.org/10.1016/j.jde.2011.08.050
  • [13] 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 (2012m:65336), https://doi.org/10.1007/s10543-011-0344-2
  • [14] 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
  • [15] R. Kruse, Strong and Weak Approximation of Semilinear Stochastic Evolution Equations, PhD thesis, University of Bielefeld, 2012.
  • [16] 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, https://doi.org/10.1093/imanum/drs055
  • [17] 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, https://doi.org/10.1214/EJP.v17-2240
  • [18] Jorge A. León and David Nualart, Stochastic evolution equations with random generators, Ann. Probab. 26 (1998), no. 1, 149-186. MR 1617045 (99b:60093), https://doi.org/10.1214/aop/1022855415
  • [19] 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, https://doi.org/10.1007/s11118-012-9276-y
  • [20] 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 (2010d:46103)
  • [21] David Nualart, The Malliavin Calculus and Related Topics, 2nd ed., Probability and its Applications (New York), Springer-Verlag, Berlin, 2006. MR 2200233 (2006j:60004)
  • [22] Vidar Thomée, Galerkin Finite Element Methods for Parabolic Problems, 2nd ed., Springer Series in Computational Mathematics, vol. 25, Springer-Verlag, Berlin, 2006. MR 2249024 (2007b:65003)
  • [23] J. M. A. M. van Neerven, Stochastic Evolution Equations, ISEM lecture notes (2008).
  • [24] 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, https://doi.org/10.1016/j.jmaa.2012.08.038

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Additional Information

Adam Andersson
Affiliation: Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, SE-412 96 Gothenburg, Sweden
Email: adam.andersson@chalmers.se

Stig Larsson
Affiliation: Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, SE-412 96 Gothenburg, Sweden
Email: stig@chalmers.se

DOI: https://doi.org/10.1090/mcom/3016
Keywords: Nonlinear stochastic heat equation, SPDE, finite element, error estimate, weak convergence, multiplicative noise, Malliavin calculus
Received by editor(s): July 26, 2013
Received by editor(s) in revised form: September 18, 2013, and November 18, 2014
Published electronically: August 20, 2015
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society