Weak convergence of Galerkin approximations for fractional elliptic stochastic PDEs with spatial white noise

TL;DR

This paper analyzes the weak convergence of Galerkin finite element methods for fractional elliptic stochastic PDEs driven by spatial white noise, providing explicit convergence rates and validating results through numerical experiments.

Contribution

It introduces a novel analysis of weak error convergence rates for Galerkin approximations of fractional elliptic SPDEs with white noise, including explicit error bounds.

Findings

Weak convergence rate is explicitly derived for the approximation.

The stochastic component's convergence rate is doubled compared to the strong rate.

Numerical experiments confirm the theoretical convergence rates.

Abstract

The numerical approximation of the solution to a stochastic partial differential equation with additive spatial white noise on a bounded domain is considered. The differential operator is assumed to be a fractional power of an integer order elliptic differential operator. The solution is approximated by means of a finite element discretization in space and a quadrature approximation of an integral representation of the fractional inverse from the Dunford-Taylor calculus. For the resulting approximation, a concise analysis of the weak error is performed. Specifically, for the class of twice continuously Fr\'echet differentiable functionals with second derivatives of polynomial growth, an explicit rate of weak convergence is derived, and it is shown that the component of the convergence rate stemming from the stochasticity is doubled compared to the corresponding strong rate. Numerical experiments for different functionals validate the theoretical results.