Strong Norm Error Bounds for Quasilinear Wave Equations Under Weak CFL-Type Conditions

In the present paper, we consider a class of quasilinear wave equations on a smooth, bounded domain. We discretize it in space with isoparametric finite elements and apply a semi-implicit Euler and midpoint rule as well as the exponential Euler and midpoint rule to obtain four fully discrete schemes. We derive rigorous error bounds of optimal order for the semi-discretization in space and the fully discrete methods in norms which are stronger than the classical $H^1 \times L^2$ energy norm under weak CFL-type conditions. To confirm our theoretical findings, we also present numerical experiments.