Optimal strong convergence rates of some Euler-type timestepping schemes for the finite element discretization SPDEs driven by additive fractional Brownian motion and Poisson random measure

TL;DR

This paper develops and analyzes finite element and Euler-type schemes for semilinear SPDEs driven by fractional Brownian motion and Poisson noise, achieving optimal convergence rates under realistic conditions.

Contribution

It introduces new numerical schemes for non-self-adjoint operators in SPDEs driven by fBm and Poisson noise, with proven optimal strong convergence rates.

Findings

Optimal convergence rate of order O(h^2 + Δt) for exponential integrator and implicit schemes.

Convergence rates depend on noise regularity and initial data.

Numerical experiments confirm theoretical results.

Abstract

In this paper, we study the numerical approximation of a general second order semilinear stochastic partial differential equation (SPDE) driven by a additive fractional Brownian motion (fBm) with Hurst parameter $H>\frac 12$ and Poisson random measure, more realistic in modelling real world phenomena. To the best of our knowledge, numerical schemes for such SPDE have been lacked in scientific literature. The approximation is done with the standard finite element method in space and three Euler-type timestepping methods in time, more precisely linear implicit method, exponential integrator and exponential Rosenbrock scheme are used for time discretisation. In contract to the current literature in the field for SPDE driven only by fBm, our linear operator is not necessary self-adjoint and optimal strong convergence rates have been achieved for SPDE driven only by fBm and SPDE driven by fBm and Poisson measure. The results examine how the convergence orders depend on the regularity of the noise and the initial data and reveal that the full discretization attains an optimal convergence rate of order $\mathcal{O}(h^2+\Delta t)$ for the exponential integrator and implicit schemes (linear operator $A$ self-adjoint for implicit). Numerical experiments are provided to illustrate our theoretical results for the case of SPDE driven with fBm noise.