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

Piecewise linear interpolation of noise in finite element approximations of parabolic SPDEs