An efficient numerical approach for stochastic evolution PDEs driven by random diffusion coefficients and multiplicative noise

Xiao Qi; Mejdi Azaiez; Can Huang; Chuanju Xu; Xiao Qi; Mejdi Azaiez; Can Huang; Chuanju Xu

doi:10.3934/math.20221134

AIMS Mathematics

2022, Volume 7, Issue 12: 20684-20710. doi: 10.3934/math.20221134

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An efficient numerical approach for stochastic evolution PDEs driven by random diffusion coefficients and multiplicative noise

1.
School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High Performance Scientific Computing, Xiamen University, 361005 Xiamen, China
2.
Bordeaux INP, Laboratoire I2M UMR 5295, 33607 Pessac, France

Received: 18 July 2022 Revised: 15 September 2022 Accepted: 16 September 2022 Published: 26 September 2022
MSC : 60H15, 60H35, 65C50

In this paper, we investigate the stochastic evolution equations (SEEs) driven by a bounded $ \log $-Whittle-Mat$ \acute{{\mathrm{e}}} $rn (W-M) random diffusion coefficient field and $ Q $-Wiener multiplicative force noise. First, the well-posedness of the underlying equations is established by proving the existence, uniqueness, and stability of the mild solution. A sampling approach called approximation circulant embedding with padding is proposed to sample the random coefficient field. Then a spatio-temporal discretization method based on semi-implicit Euler-Maruyama scheme and finite element method is constructed and analyzed. An estimate for the strong convergence rate is derived. Numerical experiments are finally reported to confirm the theoretical result.
- SEEs,
- random coefficient,
- $ Q $-Wiener multiplicative noise,
- strong convergence
Citation: Xiao Qi, Mejdi Azaiez, Can Huang, Chuanju Xu. An efficient numerical approach for stochastic evolution PDEs driven by random diffusion coefficients and multiplicative noise[J]. AIMS Mathematics, 2022, 7(12): 20684-20710. doi: 10.3934/math.20221134

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Abstract

In this paper, we investigate the stochastic evolution equations (SEEs) driven by a bounded $ \log $-Whittle-Mat$ \acute{{\mathrm{e}}} $rn (W-M) random diffusion coefficient field and $ Q $-Wiener multiplicative force noise. First, the well-posedness of the underlying equations is established by proving the existence, uniqueness, and stability of the mild solution. A sampling approach called approximation circulant embedding with padding is proposed to sample the random coefficient field. Then a spatio-temporal discretization method based on semi-implicit Euler-Maruyama scheme and finite element method is constructed and analyzed. An estimate for the strong convergence rate is derived. Numerical experiments are finally reported to confirm the theoretical result.

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