Time fractional gradient flows: Theory and numerics

TL;DR

This paper develops the theory and numerical analysis of fractional gradient flows, which are evolution equations involving memory effects characterized by Caputo derivatives, including existence, uniqueness, regularity, and error estimates.

Contribution

It introduces the concept of energy solutions for fractional gradient flows and provides the first comprehensive existence, uniqueness, regularity, and error analysis for these problems.

Findings

Established existence and uniqueness of energy solutions.

Derived reliable a posteriori error estimates.

Provided an a priori error analysis without smoothness assumptions.

Abstract

We develop the theory of fractional gradient flows: an evolution aimed at the minimization of a convex, l.s.c.~energy, with 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 the notion of energy solutions, for which we provide existence, uniqueness and certain regularizing effects. We also consider Lipschitz perturbations of this energy. For these problems we provide an a posteriori error estimate and show its reliability. This estimate depends only on the problem data, and imposes no constraints between consecutive time-steps. On the basis of this estimate we provide an a priori error analysis that makes no assumptions on the smoothness of the solution.