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MiS Preprint
60/2001
Time-space discretization of the nonlinear hyperbolic system $u_{tt}$ = div(σ(Du) + $Du_t$)
Carsten Carstensen and Georg Dolzmann
Abstract
The numerical treatment of the hyperbolic system of nonlinear wave equations with linear viscosity, $u_{ii}=div (\sigma (Du) + Du_i)$, is studied for a large class of globally Lipschitz continuous functions $\sigma$, including non-monotone stress-strain relations. The analyzed method combines an implicit Euler scheme in time with Courant (continuous and piecewise affine) finite elements in space for general time steps with varying meshes. Explicit a priori error bounds in $L^\infty (L^2)$, $L^2(W^{1,2})$, and $W^{1,2} (L^2)$ are established for the solutions of the fully discrete scheme.
