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：一般区域上随机偏微分方程的数值模拟需要对噪声进行离散化。本文在凸多面体区域上对半线性随机反应-平流-扩散方程的全离散有限元近似中采用噪声的分段线性插值。高斯噪声在时间上为白噪声，具有空间相关性，并被建模为再生核希尔伯特空间上的标准柱形维纳过程。本文首次对一般空间协方差核的噪声离散化误差进行了严格分析。假设该核定义在较大的正则区域上，从而可通过循环嵌入方法进行采样。在温和核假设下推导误差界需要非平凡的技术，包括有限元插值算子乘积的希尔伯特-施密特范数界、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误差界。数值模拟中采用FEniCS有限元软件展示了应用场景中常见核函数的算例。关键发现包括：对于d≥2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；在较粗网格上生成噪声并不总是降低精度。