Time-Space Discretization of the Nonlinear Hyperbolic System \protect\mbox\boldmath\lowercase$u_tt = \operatornamediv (\sigma(\mbox\boldmath\uppercase$D$u)+ \mbox\boldmath\uppercase$D$u_t)$

The numerical treatment of the hyperbolic system of nonlinear wave equations with linear viscosity, $u_{tt}= {\rm div}(\sigma(Du)+Du_t)$, is studied for a large class of globally Lipschitz continuous functions $\sigma$, including nonmonotone stress-strain relations. The analyzed method combines an implicit Euler scheme in time with Courant (continuous and piecewise affine) finite elements in space for a class of varying time steps with varying meshes. Explicit a priori error bounds in $L^\infty(L^2)$, $L^2(W^{1,2})$, and W1,2(L2) are established for the solutions of the fully discrete scheme.