  The convolution quadrature method originally developed for the
Riemann-Liouville fractional calculus is extended in this work to the Hadamard
fractional calculus by using the exponential type meshes. Local truncation
error analysis is presented for singular solutions. By adopting the fractional
BDF-$p(1\leq p \leq 6)$ for the Caputo-Hadamard fractional derivative in
solving subdiffusion problem with singular source terms, and using the finite
element method to discretize the space variable, we carry out the sharp error
analysis rigorously and obtain the optimal accuracy by the novel correction
technique. Our correction method is a natural generalization of the one
developed for subdiffusion problems with smooth source terms. Numerical tests
confirm the correctness of our theoretical results.
