We consider a fully discretized numerical scheme for parabolic stochastic partial differential equations with multiplicative noise. Our abstract framework can be applied to formulate a non-iterative domain decomposition approach. Such methods can help to parallelize the code and therefore lead to a more efficient implementation. The domain decomposition is integrated through the Douglas-Rachford splitting scheme, where one split operator acts on one part of the domain. For an efficient space discretization of the underlying equation, we chose the discontinuous Galerkin method as this suits the parallelization strategy well. For this fully discretized scheme, we provide a strong space-time convergence result. We conclude the manuscript with numerical experiments validating our theoretical findings.

翻译：本文研究一类具有乘性噪声的抛物型随机偏微分方程的完全离散数值格式。所提出的抽象框架可用于构建非迭代型区域分解方法，此类方法有助于实现代码并行化，从而提升计算效率。区域分解通过Douglas-Rachford分裂格式实现，其中每个分裂算子作用于区域的一个子域。针对基础方程的高效空间离散，我们选用间断伽辽金方法，因其与并行化策略高度契合。对此完全离散格式，我们给出了强意义的时空收敛性结果。最后通过数值实验验证了理论结论。