We consider a so-called fractional gradient flow: an evolution equation aimed at the minimization of a convex and l.s.c. energy, but where the
evolution has memory effects. This memory is characterized by the fact that the negative of the (sub)gradient of the energy equals the so- called Caputo derivative of the state.

We introduce a notion of “energy solutions” for which we refine the proofs of existence, uniqueness, and certain regularizing effects provided in [Li and Liu, SINUM 2019]. This is done by generalizing, to non-uniform time steps the “deconvolution” schemes of [Li and Liu, SINUM 2019], and developing a sort of “fractional minimizing movements” scheme.

We provide an a priori error estimate that seems optimal in light of the regularizing effects proved above. We also develop an a posteriori
error estimate, in the spirit of [Nochetto, Savare, Verdi, CPAM 2000] and show its reliability.

This is ongoing work with Wenbo Li (UTK).

Organizer: A. Saibaba