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

• DOI: 10.1007/s002110200394
• Publication Date: 2003-11-24T16:11:15Z
• Authors: B�atrice Rivi�re; Simon Shaw; Mary F. Wheeler; J.R. Whiteman
• Keywords: Quadrature (astronomy)

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 ? k polynomial basis on simplicies, or tensor product polynomials, ? k , on quadrilaterals). When this is not the case (e.g. using ? k 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.