Session S20 - Applied Math and Computational Methods and Analysis across the Americas

Monday, July 19, 16:30 ~ 17:00 UTC-3

Fractional Gradient Flows

Abner Salgado

Department of Mathematics, University of Tennessee, Knoxville, USA   -    This email address is being protected from spambots. You need JavaScript enabled to view it.

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

Joint work with Wenbo Li (Department of Mathematics, University of Tennessee, Knoxville).

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