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

TL;DR

This paper analyzes the error introduced by piecewise linear interpolation of noise in finite element methods for semilinear stochastic reaction-advection-diffusion equations, providing rigorous bounds and practical insights.

Contribution

It offers the first rigorous analysis of noise discretization error for general spatial covariance kernels in finite element approximations of SPDEs.

Findings

Noise interpolation does not add errors for Matérn kernels in dimensions ≥2

Some kernels produce dominant interpolation errors

Coarser mesh noise generation can still maintain accuracy

Abstract

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.