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 \(H>\frac {1}{2}\) and Poisson random measure. Such equations are more realistic in modelling real world phenomena. To the best of our knowledge, numerical schemes for such SPDE have been lacked in the scientific literature. The approximation is done with the standard finite element method in space and three Euler-type timestepping methods in time. More precisely the well-known linear implicit method, an exponential integrator and the exponential Rosenbrock scheme are used for time discretization. In contract to the current literature in the field, our linear operator is not necessary self-adjoint and we have achieved optimal strong convergence rates for 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 the optimal convergence rates of order \(\mathcal {O}(h^{2}+\varDelta t)\) for the exponential integrator and implicit schemes. Numerical experiments are provided to illustrate our theoretical results for the case of SPDE driven by the fBm noise.

在本文中，我们研究由加性分数布朗运动（fBm）和Hurst参数\（H> \ frac {1} {2} \）驱动的一般二阶半线性随机偏微分方程（SPDE）的数值逼近和泊松随机测度。在对现实世界中的现象进行建模时，此类方程式更为现实。据我们所知，科学文献中缺少这种SPDE的数值方案。在空间上使用标准有限元方法，在时间上使用三种Euler型时步法进行近似。更准确地说，使用众所周知的线性隐式方法，指数积分器和指数Rosenbrock方案进行时间离散化。与该领域的现有文献相抵触，我们的线性算子不是必需的自伴，并且我们已经通过fBm和Poisson测度实现了SPDE的最佳强收敛速度。\（\ mathcal {O}（h ^ {2} + \ varDelta t）\）用于指数积分器和隐式方案。提供数值实验来说明我们在fBm噪声驱动下的SPDE情况下的理论结果。