Weak Convergence Analysis for the Finite Element Approximation to Stochastic Allen-Cahn Equation Driven by Multiplicative White Noise
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866916660404813824 |
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| author | Zhang, Minxing Zou, Yongkui Zhang, Ran Cao, Yanzhao |
| author_facet | Zhang, Minxing Zou, Yongkui Zhang, Ran Cao, Yanzhao |
| contents | In this paper, we aim to study the optimal weak convergence order for the finite element approximation to a stochastic Allen-Cahn equation driven by multiplicative white noise. We first construct an auxiliary equation based on the splitting-up technique and derive prior estimates for the corresponding Kolmogorov equation and obtain the strong convergence order of 1 in time between the auxiliary and exact solutions. Then, we prove the optimal weak convergence order of the finite element approximation to the stochastic Allen-Cahn equation by deriving the weak convergence order between the finite element approximation and the auxiliary solution via the theory of Kolmogorov equation and Malliavin calculus. Finally, we present a numerical experiment to illustrate the theoretical analysis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_17981 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Weak Convergence Analysis for the Finite Element Approximation to Stochastic Allen-Cahn Equation Driven by Multiplicative White Noise Zhang, Minxing Zou, Yongkui Zhang, Ran Cao, Yanzhao Numerical Analysis Probability 60H15, 60H35, 65C30 G.1; G.3 In this paper, we aim to study the optimal weak convergence order for the finite element approximation to a stochastic Allen-Cahn equation driven by multiplicative white noise. We first construct an auxiliary equation based on the splitting-up technique and derive prior estimates for the corresponding Kolmogorov equation and obtain the strong convergence order of 1 in time between the auxiliary and exact solutions. Then, we prove the optimal weak convergence order of the finite element approximation to the stochastic Allen-Cahn equation by deriving the weak convergence order between the finite element approximation and the auxiliary solution via the theory of Kolmogorov equation and Malliavin calculus. Finally, we present a numerical experiment to illustrate the theoretical analysis. |
| title | Weak Convergence Analysis for the Finite Element Approximation to Stochastic Allen-Cahn Equation Driven by Multiplicative White Noise |
| topic | Numerical Analysis Probability 60H15, 60H35, 65C30 G.1; G.3 |
| url | https://arxiv.org/abs/2503.17981 |