Time fractional gradient flows, theory and numerics
A talk in the Nonlocal Equations: Analysis and Numerics series by
Abner J. Salgado from Tennessee
| Abstract: | We consider a so-called time 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). Within the CRC this talk is associated to the project(s): A7 |