  Super- and sub- diffusions are two typical types of anomalous diffusions in
the natural world. In this work, we discuss the numerical scheme for the model
describing the competition between super- and sub- diffusions driven by
fractional Brownian sheet noise. Based on the obtained regulization result of
the solution by using the properties of Mittag-Leffler function and the
regularized noise by Wong-Zakai approximation, we make full use of the
regularity of the solution operators to achieve optimal convergence of the
regularized solution. The spectral Galerkin method and the Mittag-Leffler Euler
integrator are respectively used to deal with the space and time operators. In
particular, by contour integral, the fast evaluation of the Mittag-Leffler
Euler integrator is realized. We provide complete error analyses, which are
verified by the numerical experiments.
