The Kuznetsov equation is a classical wave model of acoustics that incorporates quadratic gradient nonlinearities. When its strong damping vanishes, it undergoes a singular behavior change, switching from a parabolic-like to a hyperbolic quasilinear evolution. In this work, we establish for the first time the optimal error bounds for its finite element approximation as well as a semi-implicit fully discrete approximation that are robust with respect to the vanishing damping parameter. The core of the new arguments lies in devising energy estimates directly for the error equation where one can more easily exploit the polynomial structure of the nonlinearities and compensate inverse estimates with smallness conditions on the error. Numerical experiments are included to illustrate the theoretical results.

翻译：库兹涅佐夫方程是声学中的经典波动模型，包含二次梯度非线性项。当强阻尼消失时，该方程发生奇异行为转变，从抛物型拟线性演化切换至双曲型拟线性演化。本文首次建立了其有限元逼近以及半隐式全离散逼近的最优误差界，这些误差界关于消失的阻尼参数具有鲁棒性。新论证的核心在于直接对误差方程进行能量估计，能够更简便地利用非线性的多项式结构，并通过误差的小量条件补偿逆估计。文中包含数值实验以验证理论结果。