A mixed finite element method with reduced symmetry for the standard model in linear viscoelasticity

Abstract

We introduce and analyze a new mixed finite element method with reduced symmetry for the standard linear model in viscoelasticity. Following a previous approach employed for linear elastodynamics, the present problem is formulated as a second-order hyperbolic partial differential equation in which, after using the motion equation to eliminate the displacement unknown, the stress tensor remains as the main variable to be found. The resulting variational formulation is shown to be well-posed, and a class of \(\text {H}(\text {div})\)-conforming semi-discrete schemes is proved to be convergent. Then, we use the Newmark trapezoidal rule to obtain an associated fully discrete scheme, whose main convergence results are also established. Finally, numerical examples illustrating the performance of the method are reported.

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Acknowledgements

This research was partially supported by Spain’s Ministry of Economy Project MTM2017-87162-P; by CONICYT-Chile through the project AFB170001 of the PIA Program: Concurso Apoyo a Centros Científicos y Tecnológicos de Excelencia con Financiamiento Basal; and by Centro de Investigación en Ingeniería Matemática (CI\(^2\)MA), Universidad de Concepción.

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Correspondence to Salim Meddahi.

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Gatica, G.N., Márquez, A. & Meddahi, S. A mixed finite element method with reduced symmetry for the standard model in linear viscoelasticity. Calcolo 58, 11 (2021). https://doi.org/10.1007/s10092-021-00401-0

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Keywords

  • Mixed finite elements
  • Elastodynamics
  • Error estimates

Mathematics Subject Classification

  • 65N30
  • 65M12
  • 65M15
  • 74H15