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

Journal article
Authors David Bolin
Kristin Kirchner
Mihaly Kovacs
Published in Bit Numerical Mathematics
Volume 58
Issue 4
Pages 881-906
ISSN 0006-3835
Publication year 2018
Published at Department of Mathematical Sciences
Pages 881-906
Language en
Links dx.doi.org/10.1007/s10543-018-0719-...
Keywords Stochastic partial differential equations, Weak convergence, Gaussian white noise, Fractional, equation, fields
Subject categories Mathematics

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 Frechet 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.

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