Piecewise Linear Interpolation of Noise in Finite Element Approximations of Parabolic SPDEs

Abstract

SIAM Journal on Numerical Analysis, Volume 63, Issue 2, Page 542-563, April 2025.
Abstract. Efficient simulation of stochastic partial differential equations (SPDEs) 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 nontrivial 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 the following: noise interpolation does not introduce additional errors for Matérn kernels in [math]; there exist kernels that yield dominant interpolation errors; and generating noise on a coarser mesh does not always compromise accuracy.