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

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 Fréchet 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.

arXiv comment22 pages, 1 figure

References in corpus (5)

• Spatial models generated by nested stochastic partial differential equations, with an application to global ozone mapping
• Numerical solution of fractional elliptic stochastic PDEs with spatial white noise
• The rational SPDE approach for Gaussian random fields with general smoothness
• Weak convergence of a fully discrete approximation of a linear stochastic evolution equation with a positive-type memory term
• Spatial Matérn fields driven by non-Gaussian noise

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