TY - JOUR

T1 - Weak convergence of Galerkin approximations for fractional elliptic stochastic PDEs with spatial white noise

AU - Bolin, David

AU - Kirchner, Kristin

AU - Kovács, Mihály

N1 - Generated from Scopus record by KAUST IRTS on 2020-05-04

PY - 2018/12/1

Y1 - 2018/12/1

N2 - 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.

AB - 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.

UR - http://link.springer.com/10.1007/s10543-018-0719-8

UR - http://www.scopus.com/inward/record.url?scp=85051748005&partnerID=8YFLogxK

U2 - 10.1007/s10543-018-0719-8

DO - 10.1007/s10543-018-0719-8

M3 - Article

VL - 58

SP - 881

EP - 906

JO - BIT Numerical Mathematics

JF - BIT Numerical Mathematics

SN - 1572-9125

IS - 4

ER -