English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We consider linear scalar wave equations with a hereditary integral term ofthe kind used to model viscoelastic solids. The kernel in this Volterraintegral is a sum of decaying exponentials (The so-called Maxwell, or Zenermodel) and this allows the introduction of one of two types of families ofinternal variables, each of which evolve according to an ordinary differentialequation (ODE). There is one such ODE for each decaying exponential, and theintroduction of these ODEs means that the Volterra integral can be removed fromthe governing equation. The two types of internal variable are distinguished bywhether the unknown appears in the Volterra integral, or whether its timederivative appears; we call the resulting problems the displacement andvelocity forms. We define fully discrete formulations for each of these formsby using continuous Galerkin finite element approximations in space and animplicit `Crank-Nicolson' type of finite difference method in time. We provestability and a priori bounds, and (using the FEniCS environment,https://fenicsproject.org/) give some numerical results. These bounds do notrequire Gr\"onwall's inequality and so can be regarded to be of high quality,allowing confidence in long time integration without an a priori exponentialbuild up of error. As far as we are aware this is the first time that these twoformulations have been described together with accompanying proofs of such highquality stability and error bounds. The extension of the results tovector-valued viscoelasticity problems is straightforward and summarised at theend. The numerical results are reproducible by acquiring the python sourcesfrom https://github.com/Yongseok7717, or by running a custom built dockercontainer (instructions are given).

Finite Element Approximation and Analysis of Viscoelastic Scalar Wave Propagation with Internal Variable Formulations