  We introduce a novel discretization technique for both elliptic and parabolic
fractional diffusion problems based on double exponential quadrature formulas
and the Riesz-Dunford functional calculus. Compared to related schemes, the new
method provides faster convergence with fewer parameters that need to be
adjusted to the problem. The scheme takes advantage of any additional
smoothness in the problem without requiring a-priori knowledge to tune
parameters appropriately. We prove rigorous convergence results for both, the
case of finite regularity data as well as for data in certain Gevrey-type
classes. We confirm our findings with numerical tests.
