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Volume 36, Issue 1
Optimal Error Estimates of a Discontinuous Galerkin Method for Stochastic Allen-Cahn Equation Driven by Multiplicative Noise

Xu Yang, Weidong Zhao & Wenju Zhao

DOI: 10.4208/cicp.OA-2023-0280

Commun. Comput. Phys., 36 (2024), pp. 133-159.

Published online: 2024-07

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  • Abstract

In this paper, we develop and analyze an efficient discontinuous Galerkin method for stochastic Allen-Cahn equation driven by multiplicative noise. The proposed method is realized by symmetric interior penalty discontinuous Galerkin finite element method for space domain and implicit Euler method for time domain. Several new estimates and techniques are developed. Under some suitable regularity assumptions, we rigorously establish strong convergence results for the proposed fully discrete numerical scheme and obtain optimal convergence rates in both space and time. Numerical experiments are also carried out to validate our theoretical results and demonstrate the effectiveness of the proposed method.

  • Keywords

Stochastic Allen-Cahn equation, strong convergence, discontinuous Galerkin method, variational solution, multiplicative noise.

  • AMS Subject Headings

60H15, 60H35, 65C30, 65M60

  • Copyright

COPYRIGHT: © Global Science Press

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  • TXT
@Article{CiCP-36-133, author = {Yang , XuZhao , Weidong and Zhao , Wenju}, title = {Optimal Error Estimates of a Discontinuous Galerkin Method for Stochastic Allen-Cahn Equation Driven by Multiplicative Noise}, journal = {Communications in Computational Physics}, year = {2024}, volume = {36}, number = {1}, pages = {133--159}, abstract = {

In this paper, we develop and analyze an efficient discontinuous Galerkin method for stochastic Allen-Cahn equation driven by multiplicative noise. The proposed method is realized by symmetric interior penalty discontinuous Galerkin finite element method for space domain and implicit Euler method for time domain. Several new estimates and techniques are developed. Under some suitable regularity assumptions, we rigorously establish strong convergence results for the proposed fully discrete numerical scheme and obtain optimal convergence rates in both space and time. Numerical experiments are also carried out to validate our theoretical results and demonstrate the effectiveness of the proposed method.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2023-0280}, url = {http://global-sci.org/intro/article_detail/cicp/23299.html} }
TY - JOUR T1 - Optimal Error Estimates of a Discontinuous Galerkin Method for Stochastic Allen-Cahn Equation Driven by Multiplicative Noise AU - Yang , Xu AU - Zhao , Weidong AU - Zhao , Wenju JO - Communications in Computational Physics VL - 1 SP - 133 EP - 159 PY - 2024 DA - 2024/07 SN - 36 DO - http://doi.org/10.4208/cicp.OA-2023-0280 UR - https://global-sci.org/intro/article_detail/cicp/23299.html KW - Stochastic Allen-Cahn equation, strong convergence, discontinuous Galerkin method, variational solution, multiplicative noise. AB -

In this paper, we develop and analyze an efficient discontinuous Galerkin method for stochastic Allen-Cahn equation driven by multiplicative noise. The proposed method is realized by symmetric interior penalty discontinuous Galerkin finite element method for space domain and implicit Euler method for time domain. Several new estimates and techniques are developed. Under some suitable regularity assumptions, we rigorously establish strong convergence results for the proposed fully discrete numerical scheme and obtain optimal convergence rates in both space and time. Numerical experiments are also carried out to validate our theoretical results and demonstrate the effectiveness of the proposed method.

Yang , XuZhao , Weidong and Zhao , Wenju. (2024). Optimal Error Estimates of a Discontinuous Galerkin Method for Stochastic Allen-Cahn Equation Driven by Multiplicative Noise. Communications in Computational Physics. 36 (1). 133-159. doi:10.4208/cicp.OA-2023-0280
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