  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.
