  We study the spatial discretization of Westervelt's quasilinear strongly
damped wave equation by piecewise linear finite elements. Our approach employs
the Banach fixed-point theorem combined with a priori analysis of a linear wave
model with variable coefficients. Degeneracy of the semi-discrete Westervelt
equation is avoided by relying on the inverse estimates for finite element
functions and the stability and approximation properties of the interpolation
operator. In this way, we obtain optimal convergence rates in $L^2$-based
spatial norms for sufficiently small data and mesh size and an appropriate
choice of initial approximations. Numerical experiments in a setting of a 1D
channel as well as for a focused-ultrasound problem illustrate our theoretical
findings.
