A weak Galerkin method for nonlinear stochastic parabolic partial differential equations with additive noise

Hongze Zhu; Chenguang Zhou; Nana Sun; Hongze Zhu; Chenguang Zhou; Nana Sun

doi:10.3934/era.2022118

Electronic Research Archive

2022, Volume 30, Issue 6: 2321-2334. doi: 10.3934/era.2022118

Previous Article Next Article

Research article

A weak Galerkin method for nonlinear stochastic parabolic partial differential equations with additive noise

1.
School of Science, Shandong Jianzhu University, Jinan 250101, Shandong, China
2.
Faculty of Science, Beijing University of Technology, Beijing 100124, China
3.
Dongjia Primary School, Jinan 250101, Shandong, China

Received: 22 August 2021 Revised: 17 February 2022 Accepted: 17 February 2022 Published: 21 April 2022

In this paper, a weak Galerkin (WG for short) finite element method is used to approximate nonlinear stochastic parabolic partial differential equations with spatiotemporal additive noises. We set up a semi-discrete WG scheme for the stochastic equations, and derive the optimal order for error estimates in the sense of strong convergence.
- weak Galerkin finite element method,
- Lipschitz continuity,
- nonlinear stochastic partial differential equation,
- $ Q $-Wiener process,
- additive noise
Citation: Hongze Zhu, Chenguang Zhou, Nana Sun. A weak Galerkin method for nonlinear stochastic parabolic partial differential equations with additive noise[J]. Electronic Research Archive, 2022, 30(6): 2321-2334. doi: 10.3934/era.2022118

Related Papers:

Abstract

In this paper, a weak Galerkin (WG for short) finite element method is used to approximate nonlinear stochastic parabolic partial differential equations with spatiotemporal additive noises. We set up a semi-discrete WG scheme for the stochastic equations, and derive the optimal order for error estimates in the sense of strong convergence.

References

[1]	L. Arnold, R. Lefever (eds.), Stochastic nonlinear systems in physics, chemistry, and biology, vol. 8 of Springer Series in Synergetics, Springer-Verlag, Berlin-New York, 1981.
[2]	M. Baccouch, A finite difference method for stochastic nonlinear second-order boundary-value problems driven by additive noises, Int. J. Numer. Anal. Model., 17 (2020), 368–389.
[3]	M. Cai, S. Gan, X. Wang, Weak convergence rates for an explicit full-discretization of stochastic Allen-Cahn equation with additive noise, J. Sci. Comput., 86 (2021), Paper No. 34, 30. https://doi.org/10.1007/s10915-020-01378-8 doi: 10.1007/s10915-020-01378-8
[4]	M. Baccouch, H. Temimi, M. Ben-Romdhane, The discontinuous Galerkin method for stochastic differential equations driven by additive noises, Appl. Numer. Math., 152 (2020), 285–309. https://doi.org/10.1016/j.apnum.2019.11.020 doi: 10.1016/j.apnum.2019.11.020
[5]	R. Qi, X. Wang, Optimal error estimates of Galerkin finite element methods for stochastic Allen-Cahn equation with additive noise, J. Sci. Comput., 80 (2019), 1171–1194. https://doi.org/10.1007/s10915-019-00973-8 doi: 10.1007/s10915-019-00973-8
[6]	M. Baccouch, A stochastic local discontinuous Galerkin method for stochastic two-point boundary-value problems driven by additive noises, Appl. Numer. Math., 128 (2018), 43–64. https://doi.org/10.1016/j.apnum.2018.01.023 doi: 10.1016/j.apnum.2018.01.023
[7]	S. Chai, Y. Cao, Y. Zou, W. Zhao, Conforming finite element methods for the stochastic Cahn-Hilliard-Cook equation, Appl. Numer. Math., 124 (2018), 44–56. https://doi.org/10.1016/j.apnum.2017.09.010 doi: 10.1016/j.apnum.2017.09.010
[8]	H. Zhu, Y. Zou, S. Chai, C. Zhou, Numerical approximation to a stochastic parabolic PDE with weak Galerkin method, Numer. Math. Theory Methods Appl., 11 (2018), 604–617. https://doi.org/10.4208/nmtma.2017-oa-0122 doi: 10.4208/nmtma.2017-oa-0122
[9]	H. Zhu, Y. Zou, S. Chai, C. Zhou, A weak Galerkin method with RT elements for a stochastic parabolic differential equation, East Asian J. Appl. Math., 9 (2019), 818–830. https://doi.org/10.4208/eajam.290518.020219 doi: 10.4208/eajam.290518.020219
[10]	J. Wang, X. Ye, A weak Galerkin finite element method for second-order elliptic problems, J. Comput. Appl. Math., 241 (2013), 103–115. https://doi.org/10.1016/j.cam.2012.10.003 doi: 10.1016/j.cam.2012.10.003
[11]	M. Cui, S. Zhang, On the uniform convergence of the weak Galerkin finite element method for a singularly-perturbed biharmonic equation, J. Sci. Comput., 82 (2020), Paper No. 5, 15. https://doi.org/10.1007/s10915-019-01120-z doi: 10.1007/s10915-019-01120-z
[12]	Y. Yan, Semidiscrete Galerkin approximation for a linear stochastic parabolic partial differential equation driven by an additive noise, BIT, 44 (2004), 829–847. https://doi.org/10.1007/s10543-004-3755-5 doi: 10.1007/s10543-004-3755-5
[13]	X. Wang, Y. Zou, Q. Zhai, An effective implementation for Stokes equation by the weak Galerkin finite element method, J. Comput. Appl. Math., 370 (2020), 112586, 8. https://doi.org/10.1016/j.cam.2019.112586 doi: 10.1016/j.cam.2019.112586
[14]	J. Zhang, C. Zhou, Y. Cao, A. J. Meir, A locking free numerical approximation for quasilinear poroelasticity problems, Comput. Math. Appl., 80 (2020), 1538–1554. https://doi.org/10.1016/j.camwa.2020.07.011 doi: 10.1016/j.camwa.2020.07.011
[15]	S. Chai, Y. Zou, C. Zhou, W. Zhao, Weak Galerkin finite element methods for a fourth order parabolic equation, Numer. Methods Partial Differential Equations, 35 (2019), 1745–1755. https://doi.org/10.1002/num.22373 doi: 10.1002/num.22373
[16]	C. Zhou, Y. Zou, S. Chai, Q. Zhang, H. Zhu, Weak Galerkin mixed finite element method for heat equation, Appl. Numer. Math., 123 (2018), 180–199. https://doi.org/10.1016/j.apnum.2017.08.009 doi: 10.1016/j.apnum.2017.08.009
[17]	R. A. Adams, Sobolev spaces, Pure and Applied Mathematics, Vol. 65, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975.
[18]	G. Da Prato, J. Zabczyk, Stochastic equations in infinite dimensions, vol. 152 of Encyclopedia of Mathematics and its Applications, 2nd edition, Cambridge University Press, Cambridge, 2014. https://doi.org/10.1017/CBO9781107295513
[19]	V. Thomée, Galerkin finite element methods for parabolic problems, vol. 1054 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1984.
[20]	R. Kruse, Optimal error estimates of Galerkin finite element methods for stochastic partial differential equations with multiplicative noise, IMA J. Numer. Anal., 34 (2014), 217–251. https://doi.org/10.1093/imanum/drs055 doi: 10.1093/imanum/drs055
[21]	L. Mu, J. Wang, X. Ye, Weak Galerkin finite element methods on polytopal meshes, Int. J. Numer. Anal. Model., 12 (2015), 31–53. https://doi.org/10.1007/s10915-014-9964-4 doi: 10.1007/s10915-014-9964-4
[22]	C. Wang, J. Wang, R. Wang, R. Zhang, A locking-free weak Galerkin finite element method for elasticity problems in the primal formulation, J. Comput. Appl. Math., 307 (2016), 346–366. https://doi.org/10.1016/j.cam.2015.12.015 doi: 10.1016/j.cam.2015.12.015

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)