WIAS Preprint No. 3033, (2023)

On time-splitting methods for gradient flows with two dissipation mechanisms


Authors

  • Mielke, Alexander
    ORCID: 0000-0002-4583-3888
  • Rossi, Riccarda
    ORCID: 0000-0002-7808-0261
  • Stephan, Artur
    ORCID: 0000-0001-9871-3946

2020 Mathematics Subject Classification

  • 47J30 49J40 49J45 49J52 74D10

Keywords

  • Generalized gradient systems, energy-dissipation principle, time splitting, alternating minimizing movement scheme, repetition operators, abstract chain rule, quantiative Young estimate

DOI

10.20347/WIAS.PREPRINT.3033

Abstract

We consider generalized gradient systems in Banach spaces whose evolutions are generated by the interplay between an energy functional and a dissipation potential. We focus on the case in which the dual dissipation potential is given by a sum of two functionals and show that solutions of the associated gradient-flow evolution equation with combined dissipation can be constructed by a split-step method, i.e. by solving alternately the gradient systems featuring only one of the dissipation potentials and concatenating the corresponding trajectories. Thereby the construction of solutions is provided either by semiflows, on the time-continuous level, or by using Alternating Minimizing Movements in the time-discrete setting. In both cases the convergence analysis relies on the energy-dissipation principle for gradient systems.

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