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

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