Numerical solution of fractional elliptic stochastic PDEs with spatial white noise

Mihály Kovács,David Bolin,Kristin Kirchner

doi:10.1093/imanum/dry091

Mihály Kovács, David Bolin + Show 1 more

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https://doi.org/10.1093/imanum/dry091

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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 $t_j&gt; 0$. 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 $, $\kappa&gt; 0$ 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.

Highlights

A real-valued Gaussian random field u defined on a spatial domain D ⊂ Rd is called a Gaussian Matern field if its covariance function C : D × D → R is given by 21−ν σ2 C(x1, x2) = (κ Γ(ν) x1 − x2 )ν Kν (κ ), x1, x2 ∈ D, (1.1)where · is the Euclidean norm on Rd and Γ, Kν denote the gamma function and the modified Bessel function of the second kind, respectively
This method is based on: (i) a standard finite element discretization in space, (ii) a quadrature approximation of the inverse fractional elliptic differential operator proposed by Bonito & Pasciak (2015) for deterministic equations, and (iii) an approximation of the noise term on the right-hand side, whose covariance matrix after discretization is equal to the finite element mass matrix
We have considered the fractional order equation (2.8) with Gaussian white noise in a Hilbert space setting

Summary

Introduction

A real-valued Gaussian random field u defined on a spatial domain D ⊂ Rd is called a Gaussian Matern field if its covariance function C : D × D → R is given by 21−ν σ2. We propose an explicit numerical scheme for generating samples of an approximation to the Gaussian solution process, which allows for any fractional power β > d/4 This method is based on: (i) a standard finite element discretization in space, (ii) a quadrature approximation of the inverse fractional elliptic differential operator proposed by Bonito & Pasciak (2015) for deterministic equations, and (iii) an approximation of the noise term on the right-hand side, whose covariance matrix after discretization is equal to the finite element mass matrix. If the constructed basis has certain smoothness with respect to the differential operator, it is possible to obtain explicit rates of convergence (see, e.g., Kovacs et al, 2011, addressing this kind of problem) Constructing such an orthonormal basis requires a lot of computational effort, in particular, for complicated domains and d = 2, 3.

Model problem and main result

Partition of the error and error estimates

An application and numerical examples

Conclusion