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Volume 14, Issue 4
Strong Convergence of a Fully Discrete Scheme for Multiplicative Noise Driving SPDEs with Non-Globally Lipschitz Continuous Coefficients

Xu Yang & Weidong Zhao

DOI: 10.4208/nmtma.OA-2020-0143

Numer. Math. Theor. Meth. Appl., 14 (2021), pp. 1085-1109.

Published online: 2021-09

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  • Abstract

This work investigates strong convergence of numerical schemes for nonlinear multiplicative noise driving stochastic partial differential equations under some weaker conditions imposed on the coefficients avoiding the commonly used global Lipschitz assumption in the literature. Space-time fully discrete scheme is proposed, which is performed by the finite element method in space and the implicit Euler method in time. Based on some technical lemmas including regularity properties for the exact solution of the considered problem, strong convergence analysis with sharp convergence rates for the proposed fully discrete scheme is rigorously established.

  • Keywords

Stochastic partial differential equations, strong convergence, non-global Lipschitz, finite element method, variational solution, mean square error estimate.

  • AMS Subject Headings

60H15, 60H35, 65C30, 65M60

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-14-1085, author = {Yang , Xu and Zhao , Weidong}, title = {Strong Convergence of a Fully Discrete Scheme for Multiplicative Noise Driving SPDEs with Non-Globally Lipschitz Continuous Coefficients}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2021}, volume = {14}, number = {4}, pages = {1085--1109}, abstract = {

This work investigates strong convergence of numerical schemes for nonlinear multiplicative noise driving stochastic partial differential equations under some weaker conditions imposed on the coefficients avoiding the commonly used global Lipschitz assumption in the literature. Space-time fully discrete scheme is proposed, which is performed by the finite element method in space and the implicit Euler method in time. Based on some technical lemmas including regularity properties for the exact solution of the considered problem, strong convergence analysis with sharp convergence rates for the proposed fully discrete scheme is rigorously established.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2020-0143}, url = {http://global-sci.org/intro/article_detail/nmtma/19531.html} }
TY - JOUR T1 - Strong Convergence of a Fully Discrete Scheme for Multiplicative Noise Driving SPDEs with Non-Globally Lipschitz Continuous Coefficients AU - Yang , Xu AU - Zhao , Weidong JO - Numerical Mathematics: Theory, Methods and Applications VL - 4 SP - 1085 EP - 1109 PY - 2021 DA - 2021/09 SN - 14 DO - http://doi.org/10.4208/nmtma.OA-2020-0143 UR - https://global-sci.org/intro/article_detail/nmtma/19531.html KW - Stochastic partial differential equations, strong convergence, non-global Lipschitz, finite element method, variational solution, mean square error estimate. AB -

This work investigates strong convergence of numerical schemes for nonlinear multiplicative noise driving stochastic partial differential equations under some weaker conditions imposed on the coefficients avoiding the commonly used global Lipschitz assumption in the literature. Space-time fully discrete scheme is proposed, which is performed by the finite element method in space and the implicit Euler method in time. Based on some technical lemmas including regularity properties for the exact solution of the considered problem, strong convergence analysis with sharp convergence rates for the proposed fully discrete scheme is rigorously established.

Yang , Xu and Zhao , Weidong. (2021). Strong Convergence of a Fully Discrete Scheme for Multiplicative Noise Driving SPDEs with Non-Globally Lipschitz Continuous Coefficients. Numerical Mathematics: Theory, Methods and Applications. 14 (4). 1085-1109. doi:10.4208/nmtma.OA-2020-0143
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