Weak convergence of Galerkin approximations for fractional elliptic stochastic PDEs with spatial white noise
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The numerical approximation of the solution u to a stochastic partial differential equation with additive spatial white noise on a bounded domain in R^d is considered. The differential operator is assumed to be a fractional power L^β of an integer order elliptic differential operator L, where β∈(0,1). The solution u 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. A concise weak error analysis for the resulting approximation u_h,k^Q of u is performed. Specifically, for twice continuously Fréchet differentiable real-valued functions φ with second derivatives of polynomial growth, an explicit rate of convergence for the weak error |E[φ(u)] - E[φ(u_h,k^Q)]| 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. To this end, two time-dependent stochastic processes are introduced, which at time t=1 have the same probability distribution as the exact solution u and the approximation u_h,k^Q, respectively. The weak error is then bounded by introducing a related Kolmogorov backward equation on the interval [0,1] and applying Itô calculus. Numerical experiments for different fractional orders β∈(0,1) of the operator L=κ^2-Δ, κ > 0, with homogeneous Dirichlet boundary conditions on the domain (0,1)^d in d=1,2 spatial dimensions validate the theoretical results.
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