# Numerical solution of fractional elliptic stochastic PDEs with spatial white noise

## TL;DR

This paper develops an efficient numerical method for solving fractional elliptic stochastic PDEs with spatial white noise, using integral representations, quadrature, finite element discretization, and noise approximation, with proven convergence rates.

## Contribution

It introduces a novel numerical approach combining integral representation, quadrature, and finite element methods for fractional elliptic SPDEs driven by white noise, with explicit error analysis.

## Key findings

- Numerical experiments confirm theoretical convergence rates.
- Method effectively approximates solutions in 1D, 2D, and 3D.
- Error analysis provides explicit strong mean-square convergence rates.

## 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 non-local. 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 power operator $L^{-\beta}$ is approximated by a weighted sum of non-fractional resolvents $(I + t_j^2 L)^{-1}$ 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 the theoretical results.