An exponentially convergent discretization for space-time fractional parabolic equations using $hp$-FEM

We consider a space-time fractional parabolic problem. Combining a sinc-quadrature based method for discretizing the Riesz-Dunford integral with $hp$-FEM in space yields an exponentially convergent scheme for the initial boundary value problem with homogeneous right-hand side. For the inhomogeneous problem, an $hp$-quadrature scheme is implemented. We rigorously prove exponential convergence with focus on small times $t$, proving robustness with respect to startup singularities due to data incompatibilities.