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

Abstract

Strong and weak approximation errors of a spatial finite element method are analyzed for 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 has been no essentially sharp weak convergence rate of spatial approximation for parabolic SPDEs with nonglobally Lipschitz coefficients. To investigate the weak error, we first regularize the original equation by the splitting technique and derive the regularity estimates of the corresponding regularized Kolmogorov equation. Meanwhile, we present the regularity estimate in Malliavin sense and the refined estimate of the finite element method. Combining the regularity estimates of regularized Kolmogorov equation with Malliavin integration by parts formula, the weak convergence rate is shown to be twice the strong convergence rate.

Approximation of the invariant distribution for a class of ergodic SPDEs using an explicit tamed exponential Euler scheme

“…Proposition 6.1 is a variant of existing results: see [3] and [6] for the Lipschitz case. See also [5] and [11] for the treatment of polynomial nonlinearities, for estimates with t ď T . Proposition 6.1.…”

“…For non globally Lipschitz continuous nonlinearities, the only existing result is the recent article [12], where the authors use a fully implicit scheme. Note that the literature is also limited concerning the analysis of the weak error on finite time intervals when applied to SPDEs with non-globally Lipschitz nonlinearity: see [5] where a splitting scheme is applied for the Allen-Cahn equation (cubic nonlinearity), and also [11] and [8].…”

Weak Convergence Rates for an Explicit Full-Discretization of Stochastic Allen–Cahn Equation with Additive Noise

“…But no weak rate was addressed in this paper. In fact, for weak convergence analysis in the non-globally Lipschitz setting, we are only aware of the four papers [3,7,12,15] concerning the stochastic Allen-Cahn equations and [21] for the linearized Cahn-Hilliard-Cook equation. To the best of our knowledge, the weak convergence rates of numerical method for the stochastic Cahn-Hilliard equation are absent.…”

Convergence Rate for Galerkin Approximation of the Stochastic Allen—Cahn Equations on 2D Torus

“…Next, we consider the Galerkin approximations of (1.3): for N ≥ 1, 7) with P N , the projection operators on L 2 (T 2 ), given in (2.2), ξ N := P N ξ. By [37, (2.3)] and [29, p.4] there exist constants R N given in (3.2), which diverge logarithmically as N goes to ∞, such that…”