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

Vanja Nikolić, Barbara Wohlmuth

TL;DR

This work analyzes a spatial finite element discretization of Westervelt's quasilinear, strongly damped acoustic wave equation using piecewise linear elements. It combines a Banach fixed-point framework with a priori estimates for a linearized, variable-coefficient model to establish well-posedness and an optimal $L^2$-based convergence rate for small data and mesh sizes, while ensuring non-degeneracy through inverse estimates and Scott–Zhang interpolation. The main theoretical contributions are the existence and uniqueness of the semi-discrete solution and a rigorous $O(h^{s})$ error bound, with $1< s \\le 2$, under suitable regularity and coefficient-approximation assumptions. Numerical experiments in a 1D channel and a focused-ultrasound setting corroborate the theory and illustrate practical performance for nonlinear acoustic simulations.

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.

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

TL;DR

This work analyzes a spatial finite element discretization of Westervelt's quasilinear, strongly damped acoustic wave equation using piecewise linear elements. It combines a Banach fixed-point framework with a priori estimates for a linearized, variable-coefficient model to establish well-posedness and an optimal -based convergence rate for small data and mesh sizes, while ensuring non-degeneracy through inverse estimates and Scott–Zhang interpolation. The main theoretical contributions are the existence and uniqueness of the semi-discrete solution and a rigorous error bound, with , under suitable regularity and coefficient-approximation assumptions. Numerical experiments in a 1D channel and a focused-ultrasound setting corroborate the theory and illustrate practical performance for nonlinear acoustic simulations.

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 -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.

Paper Structure

This paper contains 15 sections, 8 theorems, 93 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Theoretical preliminaries
  3. Notation
  4. Auxiliary inequalities
  5. Finite element spaces
  6. The continuous problem
  7. Finite element approximation of the linearized Westervelt equation with variable coefficients
  8. A priori estimates for the linearized Westervelt equation with variable coefficients
  9. Auxiliary results for the elliptic projection
  10. Bounds for $\theta=R_h u-u_h$
  11. A priori estimate for the linear equation
  12. Finite element approximation of Westervelt's equation
  13. Numerical results
  14. Propagation in a channel
  15. Focused ultrasound

Key Result

Theorem 3.1

\newlabelwellposedness0 KaltenbacherLasiecka_Westervelt Let $T>0$, $b, k, c^2>0$, $0<m < \frac{1}{4k}$, and $M>0$ be arbitrary. Assume that with $\varrho_T$ sufficiently small. Then there exists a unique solution $u$ of Westervelt_continuous_weak such that and such that $\Delta u,\ u_{tt},\ \nabla u_t \in L^\infty(0,T; L^2(\Omega)), \ \nabla u_{tt} \in L^2(0,T; L^2(\Omega)).$

Figures (3)

  • Figure 1: Snapshots of the reference pressure wave in a channel \newlabelfig:Snapshots0
  • Figure 2: Propagation and self-focusing of a sound wave \newlabelfig:self-focusing0
  • Figure 3: $q(u_h)$ for different $h$ \newlabelfig:QoI0

Theorems & Definitions (16)

  • Theorem 3.1
  • Theorem 4.2
  • Proof 1
  • Lemma 5.1
  • Lemma 5.2
  • Proof 2
  • Proposition 5.3
  • Proof 3
  • Remark 5.4
  • Proposition 5.5
  • ...and 6 more