# A priori error estimates for the finite element approximation of Westervelt's quasilinear acoustic wave equation

## TL;DR

This paper provides a priori error estimates for finite element approximations of Westervelt's quasilinear acoustic wave equation, demonstrating optimal convergence rates through theoretical analysis and numerical experiments.

## Contribution

It introduces a novel a priori error analysis for finite element discretization of Westervelt's equation using Banach fixed-point theorem and inverse estimates.

## Key findings

- Optimal convergence rates in $L^2$ norms for small data and mesh size.
- Numerical experiments confirm theoretical error estimates.
- Method avoids degeneracy issues in semi-discrete equations.

## Abstract

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.