  Fractional equations have become the model of choice in several applications
where heterogeneities at the microstructure result in anomalous diffusive
behavior at the macroscale. In this work we introduce a new fractional operator
characterized by a doubly-variable fractional order and possibly truncated
interactions. Under certain conditions on the model parameters and on the
regularity of the fractional order we show that the corresponding Poisson
problem is well-posed. We also introduce a finite element discretization and
describe an efficient implementation of the finite-element matrix assembly in
the case of piecewise constant fractional order. Through several numerical
tests, we illustrate the improved descriptive power of this new operator across
media interfaces. Furthermore, we present one-dimensional and two-dimensional
$h$-convergence results that show that the variable-order model has the same
convergence behavior as the constant-order model.
