Stochastic Exponential Integrators for a Finite Element Discretization of SPDEs

TL;DR

This paper introduces a novel stochastic exponential integrator scheme for finite element discretizations of semilinear parabolic SPDEs, achieving higher order accuracy and efficient computation for complex noise-driven systems.

Contribution

The paper develops a new exponential integrator tailored for finite element discretizations of SPDEs, with convergence proofs and practical implementation techniques.

Findings

Higher order convergence in mean square norm.

Effective use of Krylov and Leja methods for exponential matrix computations.

Successful application to 2D reaction-diffusion and advection-diffusion SPDEs.

Abstract

We consider the numerical approximation of general semilinear parabolic stochastic partial differential equations (SPDEs) driven by additive space-time noise. In contrast to the standard time stepping methods which uses basic increments of the noise and the approximation of the exponential function by a rational fraction, we introduce a new scheme, designed for finite elements, finite volumes or finite differences space discretization, similar to the schemes in \cite{Jentzen3,Jentzen4} for spectral methods and \cite{GTambue} for finite element methods. We use the projection operator, the smoothing effect of the positive definite self-adjoint operator and linear functionals of the noise in Fourier space to obtain higher order approximations. We consider noise that is white in time and either in $H^1$ or $H^2$ in space and give convergence proofs in the mean square $L^{2}$ norm for a diffusion reaction equation and in mean square $ H^{1}$ norm in the presence of an advection term. For the exponential integrator we rely on computing the exponential of a non-diagonal matrix. In our numerical results we use two different efficient techniques: the real fast \Leja points and Krylov subspace techniques. We present results for a linear reaction diffusion equation in two dimensions as well as a nonlinear example of two-dimensional stochastic advection diffusion reaction equation motivated from realistic porous media flow.