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Strong and weak approximation errors of a spatial finite element method areanalyzed for stochastic partial differential equations(SPDEs) with one-sidedLipschitz coefficients, including the stochastic Allen--Cahn equation, drivenby additive noise. In order to give the strong convergence rate of the finiteelement method, we present an appropriate decomposition and some a prioriestimates of the discrete stochastic convolution. To the best of our knowledge,there has been no essentially sharp weak convergence rate of spatialapproximation for parabolic SPDEs with non-globally Lipschitz coefficients. Toinvestigate the weak error, we first regularize the original equation by thesplitting technique and derive the regularity estimates of the correspondingregularized Kolmogorov equation. Meanwhile, we present the regularity estimatein Malliavin sense and the refined estimate of the finite element method.Combining the regularity estimates of regularized Kolmogorov equation withMalliavin integration by parts formula, the weak convergence rate is shown tobe twice the strong convergence rate.

Strong and weak convergence rates of finite element method for stochastic partial differential equation with non-sided Lipschitz coefficient