Weak convergence of Galerkin approximations for fractional elliptic stochastic PDEs with spatial white noise
Provides rigorous weak error analysis for a class of fractional SPDEs, benefiting numerical analysts and practitioners in computational stochastic PDEs.
The paper analyzes weak convergence of finite element approximations for fractional elliptic SPDEs with spatial white noise, deriving an explicit weak rate that doubles the strong rate for smooth functionals. Numerical experiments confirm the theoretical findings.
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.