  This paper deals with the numerical approximation of semilinear parabolic
stochastic partial differential equation (SPDE) driven simultaneously by
Gaussian noise and Poisson random measure, more realistic in modeling real
world phenomena. The SPDE is discretized in space with the standard finite
element method and in time with the linear implicit Euler method or an
exponential integrator, more efficient and stable for stiff problems. We prove
the strong convergence of the fully discrete schemes toward the mild solution.
The results reveal how convergence orders depend on the regularity of the noise
and the initial data.In addition, we exceed the classical orders $1/2$ in time
and $1$ in space achieved in the literature when dealing with SPDE driven by
Poisson measure with less regularity assumptions on the nonlinear drift
function. In particular, for trace class multiplicative Gaussian noise we
achieve convergence order $\mathcal{O}(h^2+\Delta t^{1/2})$.For additive trace
class Gaussian noise and an appropriate jump function, we achieve convergence
order $\mathcal{O}(h^2+\Delta t)$. Numerical experiments to sustain the
theoretical results are provided.
