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.

中文翻译：

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弱 CFL 型条件下拟线性波动方程的强范数误差界

在本文中，我们考虑光滑有界域上的一类拟线性波动方程。我们用等参有限元在空间中对其进行离散，并应用半隐式欧拉和中点规则以及指数欧拉和中点规则以获得四种完全离散的格式。我们推导了空间半离散化的最优阶严格误差界和规范中的全离散方法，其在弱CFL型条件下比经典\(H^1\times L^2\)能量范数更强。为了证实我们的理论发现，我们还进行了数值实验。