  Fractional vector calculus is the building block of the fractional partial
differential equations that model non-local or long-range phenomena, e.g.,
anomalous diffusion, fractional electromagnetism, and fractional
advection-dispersion. In this work, we reformulate a type of fractional vector
calculus that uses Caputo fractional partial derivatives and discretize this
reformulation using discrete exterior calculus on a cubical complex in the
structure-preserving way, meaning that the continuous-level properties
$\operatorname{curl}^\alpha \operatorname{grad}^\alpha = \mathbf{0}$ and
$\operatorname{div}^\alpha \operatorname{curl}^\alpha = 0$ hold exactly on the
discrete level. We discuss important properties of our fractional discrete
exterior derivatives and verify their second-order convergence in the root mean
square error numerically. Our proposed discretization has the potential to
provide accurate and stable numerical solutions to fractional partial
differential equations and exactly preserve fundamental physics laws on the
discrete level regardless of the mesh size.
