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

Simulation of stochastic partial differential equations (SPDE) on a general domain requires a discretization of the noise. In this paper, the noise is discretized by a piecewise linear interpolation. The error caused by this is analyzed in the context of a fully discrete finite element approximation of a semilinear stochastic reaction-advection-diffusion equation on a convex polygon. The noise is Gaussian, white in time and correlated in space. It is modeled as a standard cylindrical Wiener process on the reproducing kernel Hilbert space associated to the covariance kernel. The noise is assumed to extend to a larger polygon than the SPDE domain to allow for sampling by the circulant embedding method. The interpolation error is analyzed under mild assumptions on the kernel. The main tools used are Hilbert--Schmidt bounds of multiplication operators onto negative order Sobolev spaces and an error bound for the finite element interpolant in fractional Sobolev norms. Examples with covariance kernels encountered in applications are illustrated in numerical simulations using the FEniCS finite element software. Conclusions from the analysis include that interpolation of noise with Mat\'ern kernels does not cause an additional error, that there exist kernels where the interpolation error dominates and that generation of noise on a coarser mesh than that of the SPDE discretization does not always result in a loss of accuracy.