Numerical solution of fractional elliptic stochastic PDEs with spatial white noise

Abstract

Abstract The numerical approximation of solutions to stochastic partial differential equations with additive spatial white noise on bounded domains in $\mathbb{R}^d$ is considered. The differential operator is given by the fractional power $L^\beta $, $\beta \in (0,1)$ of an integer-order elliptic differential operator $L$ and is therefore nonlocal. Its inverse $L^{-\beta }$ is represented by a Bochner integral from the Dunford–Taylor functional calculus. By applying a quadrature formula to this integral representation the inverse fractional-order operator $L^{-\beta }$ is approximated by a weighted sum of nonfractional resolvents $( I + \exp(2 y_\ell) \, L )^{-1}$ at certain quadrature nodes $t_j> 0$. The resolvents are then discretized in space by a standard finite element method. This approach is combined with an approximation of the white noise, which is based only on the mass matrix of the finite element discretization. In this way an efficient numerical algorithm for computing samples of the approximate solution is obtained. For the resulting approximation the strong mean-square error is analyzed and an explicit rate of convergence is derived. Numerical experiments for $L=\kappa ^2-\Delta $, $\kappa> 0$ with homogeneous Dirichlet boundary conditions on the unit cube $(0,1)^d$ in $d=1,2,3$ spatial dimensions for varying $\beta \in (0,1)$ attest to the theoretical results.