The present article deals with strong approximations of additive noise driven stochastic partial differential equations (SPDEs) with nonglobally Lipschitz nonlinearity in a bounded domain $ \mathcal{D} \in{\mathbb{R}}^{d}$, $ d \leq 3$. As the first contribution, we establish the well-posedness and regularity of the considered SPDEs in space dimension $d \le 3$, under more relaxed assumptions on the stochastic convolution. This improves relevant results in the literature and covers both the space-time white noise ($d=1$) and the trace-class noises ($\text{Tr} (Q) < \infty $) in multiple dimensions $d=2,3$. Such an improvement is achieved based on a key perturbation estimate for a perturbed PDE, with the aid of which we prove the convergence and uniform regularity of a spectral approximation of the SPDEs and thus get the improved regularity results. The second contribution of the paper is to propose and analyze a spatio-temporal discretization of the SPDEs, by incorporating a standard finite element method in space and a linearly implicit nonlinearity-tamed Euler method for the temporal discretization. The proposed time-stepping scheme is linearly implicit and does not suffer from solving nonlinear algebra equations as the backward Euler scheme does. Based on the improved regularity results, we recover the expected strong convergence rates of the fully discrete scheme and reveal how the convergence rates rely on the regularity of the noise process. In particular, a classical convergence rate of order $O(h^{2} +\tau )$ can be obtained even in high dimension $d=3$, as the driven noise is of trace class and satisfies certain regularity assumptions. The optimal error estimates turn out to be challenging and face some essential difficulties when the tamed time-stepping scheme meets the finite element spatial discretization, particularly in the context of low regularity and multiple dimensions $d \le 3$. Some highly nontrivial arguments are introduced to overcome the difficulties. Finally, numerical examples corroborate the claimed strong orders of convergence.

中文翻译：

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具有非全局 Lipschitz 系数的 SPDE 的线性隐式有限元全离散化格式

本文讨论有界域中具有非全局 Lipschitz 非线性的加性噪声​​驱动随机偏微分方程 (SPDE) 的强近似 $ \mathcal{D} \in{\mathbb{R}}^{d}$, $ d \ leq 3$。作为第一个贡献，我们在随机卷积的更宽松的假设下，在空间维度 $d \le 3$ 中建立了所考虑的 SPDE 的适定性和规律性。这改进了文献中的相关结果，并涵盖了多维 $d 中的时空白噪声 ($d=1$) 和迹类噪声 ($\text{Tr} (Q) < \infty $) =2,3$。这种改进是基于扰动偏微分方程的关键扰动估计来实现的，借助它我们证明了SPDE谱近似的收敛性和一致规律性，从而得到了改进的规律性结果。本文的第二个贡献是通过结合空间中的标准有限元方法和用于时间离散化的线性隐式非线性驯服欧拉方法，提出并分析了 SPDE 的时空离散化。所提出的时间步进方案是线性隐式的，并且不会像向后欧拉方案那样求解非线性代数方程。基于改进的规律性结果，我们恢复了完全离散方案的预期强收敛速度，并揭示了收敛速度如何依赖于噪声过程的规律性。特别是，即使在高维 $d=3$ 下，也可以获得 $O(h^{2} +\tau )$ 阶的经典收敛率，因为驱动噪声是迹类并且满足某些规律性假设。当驯服的时间步长方案满足有限元空间离散化时，最优误差估计变得具有挑战性，并面临一些基本困难，特别是在低规律性和多维 $d \le 3$ 的情况下。引入一些非常重要的论点来克服困难。最后，数值例子证实了所声称的强收敛阶数。