Weak Convergence Analysis for the Finite Element Approximation to Stochastic Allen-Cahn Equation Driven by Multiplicative White Noise
By: Minxing Zhang , Yongkui Zou , Ran Zhang and more
Potential Business Impact:
Makes math models of tricky things more accurate.
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.
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