extended report for: An a priori error estimate for a temporally discontinuous Galerkin space-time finite element method for linear elasto- and visco-dynamics

Abstract

We extend the formulation and a priori error analysis given by Johnson (Discontinuous Galerkin finite element methods for second order hyperbolic problems, Comp. Meth. Appl. Mech. Eng., 107:117—129, 1993) from the acoustic wave equation to a Voigt and Maxwell-Zener viscodynamic system incorporating Rayleigh damping. The elastic term in the Rayleigh damping introduces a multiplicative T1/2 growth in the constant but otherwise the error bound is consistent with that obtained by Johnson, with a constant that grows a priori with T1/2 and also with norms of the solution. Gronwall’s inequality is not used and so we can expect that this bound is of high enough quality to afford confidence in longtime integration. The viscoelasticity is modelled by internal variables that evolve according to ordinary differential equations and so the system shares similarities with dispersive Debye and Drude metamaterial models currently being studied in electromagnetism, as well as to acoustic metamaterial systems. This appears to be the first time an a priori error analysis has been given for DG-in-time treatment of dispersive problems of this type.