  The main contribution of this work is to construct and analyze stable and
high order schemes to efficiently solve the two-dimensional time
Caputo-Fabrizio fractional diffusion equation. Based on a third-order finite
difference method in time and spectral methods in space, the proposed scheme is
unconditionally stable and has the global truncation error
$\mathcal{O}(\tau^3+N^{-m})$, where $\tau$, $N$ and $m$ are the time step size,
polynomial degree and regularity in the space variable of the exact solution,
respectively. It should be noted that the global truncation error
$\mathcal{O}(\tau^2+N^{-m})$ is well established in [ Li, Lv and Xu, {\em
Numer. Methods Partial Differ. Equ}. (2019)]. Finally, some numerical
experiments are carried out to verify the theoretical analysis. To the best of
our knowledge, this is the first proof for the stability of the third-order
scheme for the Caputo-Fabrizio fractional operator.
