Time Fractional Gradient Flows: Theory and Numerics

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 joint work with Wenbo Li (UTK).