# Numerical solution of fractional elliptic stochastic PDEs with spatial white noise Journal article, 2020

The numerical approximation of solutions to stochastic partial differential equations with additive spatial white noise on bounded domains in R-d is considered. The differential operator is given by the fractional power L-beta, beta is an element of (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(2yl)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 is an element of (0, 1) attest to the theoretical results.

spatial statistics

fractional operators

Matern covariances

stochastic partial differential equations

Gaussian white noise

finite element methods

## Author

### David Bolin

Chalmers, Mathematical Sciences, Applied Mathematics and Statistics

### Kristin Kirchner

Chalmers, Mathematical Sciences

University of Gothenburg

### Mihaly Kovacs

Chalmers, Mathematical Sciences, Applied Mathematics and Statistics

#### IMA Journal of Numerical Analysis

0272-4979 (ISSN) 1464-3642 (eISSN)

Vol. 40 2 1051-1073

### Subject Categories

Computational Mathematics

Signal Processing

Mathematical Analysis

### DOI

10.1093/imanum/dry091