Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
184 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
45 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Convergence of a spatial semidiscretization for a three-dimensional stochastic Allen-Cahn equation with multiplicative noise (2401.09834v7)

Published 18 Jan 2024 in math.NA, cs.NA, and math.PR

Abstract: This paper studies the convergence of a spatial semidiscretization of a three-dimensional stochastic Allen-Cahn equation with multiplicative noise. For non-smooth initial data, the regularity of the mild solution is investigated, and an error estimate is derived within the spatial (L2)-norm setting. In the case of smooth initial data, two error estimates are established within the framework of general spatial (Lq)-norms.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (34)
  1. Strong convergence rates for explicit space-time discrete numerical approximations of stochastic Allen-Cahn equations. Stoch PDE: Anal. Comp., 11:211–268, 2023.
  2. S. Becker and A. Jentzen. Strong convergence rates for nonlinearity-truncated Euler-type approximations of stochastic Ginzburg-Landau quations. Stoch. Process. Appl., 129:28–69, 2019.
  3. N. Berglund. An introduction to singular stochastic PDEs: Allen-Cahn equations, metastability and regularity structures. arXiv:1901.07420v2, 2019.
  4. Strong convergence rates of semidiscrete splitting approximations for the stochastic Allen-Cahn equation. IMA J. Numer. Anal., 39:2096–2134, 2019.
  5. C.-E. Bréhier and L. Goudenège. Weak convergence rates of splitting schemes for the stochastic Allen-Cahn equation. BIT Numer. Math., 60:543–582, 2020.
  6. S. C. Brenner and R. Scott. The mathematical theory of finite element methods. Springer-Verlag New York, 3 edition, 2008.
  7. Itô’s formula in UMD Banach spaces and regularity of solutions of the Zakai equation. J. Differ. Equ., 245:30–58, 2008.
  8. Finite element methods for the stochastic Allen-Cahn equation with gradient-type multiplicative noises. SIAM J. Numer. Anal., 55:194–216, 2017.
  9. T. Funaki. The scaling limit for a stochastic PDE and the separation of phases. Probab. Theory Relat. Fields, 102:221–288, 1995.
  10. Analysis in Banach spaces. Springer, Cham, 2016.
  11. Analysis in Banach spaces. Springer, Cham, 2017.
  12. A. Jentzen and P. Pušnik. Strong convergence rates for an explicit numerical approximation method for stochastic evolution equations with non-globally Lipschitz continuous nonlinearities. IMA J. Numer. Anal., 40:1005–1050, 2020.
  13. Action minimization and sharp-interface limits for the stochastic Allen-Cahn equation. Comm. Pure Appl. Math., 60:393–438, 2007.
  14. G. Leoni, editor. A first course in Sobolev spaces. American Mathematical Society, 2 edition, 2017.
  15. B. Li. Maximum-norm stability and maximal Lpsuperscript𝐿𝑝L^{p}italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT regularity of FEMs for parabolic equations with Lipschitz continuous coefficients. Numer. Math., 131:489–516, 2015.
  16. B. Li and Z. Zhou. Discrete stochastic maximal Lp-regularity and convergence of a spatial semidiscretization for a stochastic parabolic equation. arXiv:2311.04615, 2023.
  17. W. Liu. Well-posedness of stochastic partial differential equations with Lyapunov condition. J. Differ. Equ., 255:572–592, 2013.
  18. W. Liu and M. Röckner. SPDE in Hilbert space with locally monotone coefficients. J. Funct. Anal., 259:2902–2922, 2010.
  19. Z. Liu and Z. Qiao. Wong-Zakai approximation of stochastic Allen-Cahn equations. Int. J. Numer. Anal. Model., 16:681–694, 2019.
  20. Z. Liu and Z. Qiao. Strong approximation of monotone stochastic partial differential equations driven by multiplicative noise. Stoch PDE: Anal. Comp., 9:559–602, 2021.
  21. A. Lunardi. Interpolation theory. Edizioni della Normale, Pisa, 2018.
  22. A. Majee and A. Prohl. Optimal strong rates of convergence for a space-time discretization of the stochastic Allen-Cahn equation with multiplicative noise. Comput. Methods Appl. Math., 18(2):297–311, 2018.
  23. Numerical approximation of nonlinear SPDE’s. Stoch PDE: Anal. Comp., 11:1553–1634, 2023.
  24. A. Pazy. Semigroups of linear operators and applications to partial differential equations. Springer, New York, 1983.
  25. R. Qi and X. Wang. Optimal error estimates of Galerkin finite element methods for stochastic Allen-Cahn equation with additive noise. J. Sci. Comput., 80:1171–1194, 2019.
  26. An efficient approximation to the stochastic Allen-Cahn equation with random diffusion coefficient field and multiplicative noise. Adv. Comput. Math., 49:73, 2023.
  27. M. Röger and H. Weber. Tightness for a stochastic Allen–Cahn equation. Stoch PDE: Anal. Comp., 1:175–203, 2013.
  28. L. Tartar. An Introduction to Sobolev Spaces and Interpolation Spaces. Springer, Berlin, 2007.
  29. J. Thomas. Numerical Partial Differential Equations. Texts in Applied Mathematics, 2010.
  30. V. Thomée. Galerkin Finite Element Methods for Parabolic Problems. Springer, Berlin, 2006.
  31. Maximal Lpsuperscript𝐿𝑝L^{p}italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT-regularity for stochastic evolution equations. SIAM J. Math. Anal., 44:1372–1414, 2012.
  32. Stochastic maximal Lpsuperscript𝐿𝑝L^{p}italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT-regularity. Ann. Probab., 40:788–812, 2012.
  33. X. Wang. An efficient explicit full-discrete scheme for strong approximation of stochastic Allen–Cahn equation. Stoch. Process. Appl., 130:6271–6299, 2020.
  34. L. Weis. Operator-valued Fourier multiplier theorems and maximal Lpsubscript𝐿𝑝L_{p}italic_L start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT-regularity. Math. Ann., 319:735–758, 2001.
Citations (2)
View on Semantic Scholar

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now