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Discontinuous Galerkin finite element methods for linear elasticity and quasistatic linear viscoelasticity

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Summary.

We consider a finite-element-in-space, and quadrature-in-time-discretization of a compressible linear quasistatic viscoelasticity problem. The spatial discretization uses a discontinous Galerkin finite element method based on polynomials of degree r—termed DG(r)—and the time discretization uses a trapezoidal-rectangle rule approximation to the Volterra (history) integral. Both semi- and fully-discrete a priori error estimates are derived without recourse to Gronwall's inequality, and therefore the error bounds do not show exponential growth in time. Moreover, the convergence rates are optimal in both h and r providing that the finite element space contains a globally continuous interpolant to the exact solution (e.g. when using the standard ℙk polynomial basis on simplicies, or tensor product polynomials, ℚk, on quadrilaterals). When this is not the case (e.g. using ℙk on quadri-laterals) the convergence rate is suboptimal in r but remains optimal in h. We also consider a reduction of the problem to standard linear elasticity where similarly optimal a priori error estimates are derived for the DG(r) approximation.

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Authors and Affiliations

  1. Center for Subsurface Modeling, TICAM, University of Texas, Austin, TX, 78712, USA

    Béatrice Rivière & Mary F. Wheeler

  2. BICOM, Brunel University, Uxbridge, UB8 3PH, U.K

    Simon Shaw & J.R. Whiteman

Authors
  1. Béatrice Rivière
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  2. Simon Shaw
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  3. Mary F. Wheeler
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  4. J.R. Whiteman
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Corresponding author

Correspondence to J.R. Whiteman.

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Mathematics Subject Classification (2000): 65N36

Shaw and Whiteman would like to acknowledge the support of the US Army Research Office, Grant #DAAD19-00-1-0421, and the UK EPSRC, Grant #GR/R10844/01. Whiteman would also like to acknowledge support from TICAM in the form of Visiting Research Fellowships.

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Rivière, B., Shaw, S., Wheeler, M. et al. Discontinuous Galerkin finite element methods for linear elasticity and quasistatic linear viscoelasticity. Numer. Math. 95, 347–376 (2003). https://doi.org/10.1007/s002110200394

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  • DOI: https://doi.org/10.1007/s002110200394

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