Strong and Weak Convergence Rates of a Spatial Approximation for Stochastic Partial Differential Equation with One-sided Lipschitz Coefficient

Strong and weak approximation errors of a spatial finite element method are analyzed for the stochastic partial differential equations (SPDEs) with one-sided Lipschitz coefficients, including the stochastic Allen--Cahn equation, driven by additive noise. In order to give the strong convergence rate of the finite element method, we present an appropriate decomposition and some a priori estimates of the discrete stochastic convolution. To the best of our knowledge, there have been no essentially sharp weak convergence rates of spatial approximations for parabolic SPDEs with non-globally Lipschitz coefficients. To investigate the weak error, we first regularize the original equation by the splitting technique and obtain the regularity estimate the corresponding regularized Kolmogorov equation. Meanwhile, we present the refined estimates and the regularity estimate in the Malliavin sense of the finite element methods. Combining with the regularity of regularized Kolmogorov equation and Malliavin integration by parts, the weak convergence rate is shown to be twice the strong convergence rate.