  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.
