Convergence of finite element solution of stochastic Burgers equation

Jingyun Lv; Xiaoyan Lu; Jingyun Lv; Xiaoyan Lu

doi:10.3934/era.2024076

Electronic Research Archive

2024, Volume 32, Issue 3: 1663-1691. doi: 10.3934/era.2024076

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Convergence of finite element solution of stochastic Burgers equation

Jingyun Lv ^{1
,
,},
Xiaoyan Lu ²

1.
School of Fundamental Education, Beijing Polytechnic College, Beijing 100042, China
2.
Basic Courses Department of Beijing Polytechnic, Beijing 100176, China

Academic Editor: Xuehua Yang

Received: 26 November 2023 Revised: 23 January 2024 Accepted: 06 February 2024 Published: 20 February 2024

We explore the numerical approximation of the stochastic Burgers equation driven by fractional Brownian motion with Hurst index $ H\in(1/4, 1/2) $ and $ H\in(1/2, 1) $, respectively. The spatial and temporal regularity properties for the solution are obtained. The given problem is discretized in time with the implicit Euler scheme and in space with the standard finite element method. We obtain the strong convergence of semidiscrete and fully discrete schemes, performing the error estimates on a subset $ \Omega_{k, h} $ of the sample space $ \Omega $ with the Gronwall argument being used to overcome the difficulties, caused by the subtle interplay of the nonlinear convection term. Numerical examples confirm our theoretical findings.
- stochastic Burgers equation,
- fractional Brownian motion,
- finite element method,
- error estimates,
- strong convergence
Citation: Jingyun Lv, Xiaoyan Lu. Convergence of finite element solution of stochastic Burgers equation[J]. Electronic Research Archive, 2024, 32(3): 1663-1691. doi: 10.3934/era.2024076

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Abstract

We explore the numerical approximation of the stochastic Burgers equation driven by fractional Brownian motion with Hurst index $ H\in(1/4, 1/2) $ and $ H\in(1/2, 1) $, respectively. The spatial and temporal regularity properties for the solution are obtained. The given problem is discretized in time with the implicit Euler scheme and in space with the standard finite element method. We obtain the strong convergence of semidiscrete and fully discrete schemes, performing the error estimates on a subset $ \Omega_{k, h} $ of the sample space $ \Omega $ with the Gronwall argument being used to overcome the difficulties, caused by the subtle interplay of the nonlinear convection term. Numerical examples confirm our theoretical findings.

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