Generalized convolution quadrature for the fractional integral and fractional diffusion equations

We consider the application of the generalized Convolution Quadrature (gCQ) of the first order to approximate fractional integrals and associated fractional diffusion equations. The gCQ is a generalization of Lubich's Convolution Quadrature (CQ) which allows for variable steps. In this paper we analyze the application of the gCQ to fractional integrals, with a focus on the low regularity case. It is well known that in this situation the original CQ presents an order reduction close to the singularity. Moreover, the available theory for the gCQ does not cover this situation. Here we deduce error bounds for a general time mesh. We show first order of convergence under much weaker regularity requirements than previous results in the literature. We also prove that uniform first order convergence is achievable for a graded time mesh, which is appropriately refined close to the singularity, according to the order of the fractional integral and the regularity of the data. Then we study how to obtain full order of convergence for the application to fractional diffusion equations. For the implementation of this method, we use fast and oblivious quadrature and present several numerical experiments to illustrate our theoretical results.