Efficient simulation of stochastic partial differential equations (SPDE) on general domains requires noise discretization. This paper employs piecewise linear interpolation of noise in a fully discrete finite element approximation of a semilinear stochastic reaction-advection-diffusion equation on a convex polyhedral domain. The Gaussian noise is white in time, spatially correlated, and modeled as a standard cylindrical Wiener process on a reproducing kernel Hilbert space. This paper provides the first rigorous analysis of the resulting noise discretization error for a general spatial covariance kernel. The kernel is assumed to be defined on a larger regular domain, allowing for sampling by the circulant embedding method. The error bound under mild kernel assumptions requires non-trivial techniques like Hilbert--Schmidt bounds on products of finite element interpolants, entropy numbers of fractional Sobolev space embeddings and an error bound for interpolants in fractional Sobolev norms. Examples with kernels encountered in applications are illustrated in numerical simulations using the FEniCS finite element software. Key findings include: noise interpolation does not introduce additional errors for Mat\'ern kernels in $d\ge2$; there exist kernels that yield dominant interpolation errors; and generating noise on a coarser mesh does not always compromise accuracy.

翻译：在一般区域上高效模拟随机偏微分方程（SPDE）需要对噪声进行离散化。本文针对凸多面体区域上的半线性随机反应-平流-扩散方程，在全离散有限元逼近中采用分段线性噪声插值方法。该高斯噪声在时间上呈白噪声特性，在空间上具有相关性，并被建模为再生核希尔伯特空间上的标准柱面维纳过程。本文首次对一般空间协方差核所产生的噪声离散化误差进行了严格分析。该核被假定定义在更大的规则区域上，从而允许通过循环嵌入方法进行采样。在温和的核假设下，误差界的推导需要运用非平凡的技术手段，包括有限元插值算子乘积的希尔伯特-施密特界、分数阶索伯列夫空间嵌入的熵数，以及分数阶索伯列夫范数下插值算子的误差界。利用FEniCS有限元软件进行的数值模拟展示了应用中常见核的算例。主要发现包括：对于$d\ge2$维的Matérn核，噪声插值不会引入额外误差；存在某些核会导致插值误差占主导地位；在更粗网格上生成噪声并不总是降低精度。