The Gradient Flow Structure of an Extended Maxwell Viscoelastic Model and a Structure-Preserving Finite Element Scheme

  • Masato Kimura
  • Hirofumi Notsu
  • Yoshimi Tanaka
  • Hiroki Yamamoto


An extended Maxwell viscoelastic model with a relaxation parameter is studied from mathematical and numerical points of view. It is shown that the model has a gradient flow property with respect to a viscoelastic energy. Based on the gradient flow structure, a structure-preserving time-discrete model is proposed and existence of a unique solution is proved. Moreover, a structure-preserving P1/P0 finite element scheme is presented and its stability in the sense of energy is shown by using its discrete gradient flow structure. As typical viscoelastic phenomena, two-dimensional numerical examples by the proposed scheme for a creep deformation and a stress relaxation are shown and the effects of the relaxation parameter are investigated.


Gradient flow structure Maxwell viscoelastic model Finite element method Structure preserving scheme 



This work is partially supported by JSPS KAKENHI Grant Numbers JP16H02155, JP17H02857, JP26800091, JP16K13779, JP18H01135, and JP17K05609, JSPS A3 Foresight Program, and JST PRESTO Grant Number JPMJPR16EA.


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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Faculty of Mathematics and PhysicsKanazawa UniversityKanazawaJapan
  2. 2.Japan Science and Technology Agency, PRESTOKawaguchiJapan
  3. 3.Department of Environment and System Sciences, Graduate School of Environment and Information SciencesYokohama National UniversityYokohamaJapan
  4. 4.Graduate School of Natural Science and TechnologyKanazawa UniversityKanazawaJapan

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