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

TL;DR

This paper develops an a priori error analysis for a finite element and finite difference scheme applied to a fractional order viscoelasticity problem modeled by a Volterra integral equation with a weakly singular kernel, providing stability bounds and error estimates.

Contribution

It introduces a novel a priori error analysis for finite element approximation of fractional viscoelasticity problems with weakly singular kernels, including stability bounds and numerical validation.

Findings

Established stability bounds for the numerical scheme.

Derived a priori error estimates considering solution regularity.

Validated results through numerical experiments.

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.