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

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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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Funding

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

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Author notes

  1. Yongseok Jang

    Present address: CERFACS, 42 Avenue Gaspard Coriolis, 31057, Toulouse, France

Authors and Affiliations

  1. Department of Mathematics, Brunel University London, Uxbridge, UB8 3PH, UK

    Yongseok Jang & Simon Shaw

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  1. Yongseok Jang
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  2. Simon Shaw
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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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