A priori error analysis for a finite element approximation of dynamic viscoelasticity problems involving a fractional order integro-differential constitutive law

Abstract

We consider a fractional order viscoelasticity problem modelled by a power-law type stress relaxation function. This viscoelastic problem is a Volterra integral equation of the second kind with a weakly singular kernel where the convolution integral corresponds to fractional order differentiation/integration. We use a spatial finite element method and a finite difference scheme in time. Due to the weak singularity, fractional order integration in time is managed approximately by linear interpolation so that we can formulate a fully discrete problem. In this paper, we present a stability bound as well as a priori error estimates. Furthermore, we carry out numerical experiments with varying regularity of exact solutions at the end.

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Jang gratefully acknowledges the supported of a scholarship from Brunel University London.

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Correspondence to Yongseok Jang.

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Communicated by: Bangti Jin

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Jang, Y., Shaw, S. A priori error analysis for a finite element approximation of dynamic viscoelasticity problems involving a fractional order integro-differential constitutive law. Adv Comput Math 47, 46 (2021). https://doi.org/10.1007/s10444-021-09857-8

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Keywords

  • Viscoelasticity
  • Power-law
  • Fractional calculus
  • Finite element method
  • A priori error estimates

Mathematics Subject Classification 2010

  • 4D05
  • 74S05
  • 45D05