$hp$-FEM for the fractional heat equation

We consider a time dependent problem generated by a nonlocal operator in space. Applying a discretization scheme based on $hp$-Finite Elements and a Caffarelli-Silvestre extension we obtain a semidiscrete semigroup. The discretization in time is carried out by using $hp$-Discontinuous Galerkin based timestepping. We prove exponential convergence for such a method in an abstract framework for the discretization in the original domain $\Omega$.