Mixed approximation of nonlinear acoustic equations: Well-posedness and a priori error analysis

TL;DR

This paper develops mixed finite element methods for nonlinear acoustic equations, establishing well-posedness, stability, and error bounds, with numerical validation for ultrasonic wave simulations.

Contribution

It introduces a novel mixed formulation for nonlinear acoustic equations, providing rigorous analysis and error estimates, including uniform bounds for the Westervelt equation.

Findings

Proved well-posedness and stability under certain conditions.

Derived optimal a priori error estimates in energy norm.

Confirmed theoretical results through numerical experiments with RT and BDM elements.

Abstract

Accurate simulation of nonlinear acoustic waves is essential for the continued development of a wide range of (high-intensity) focused ultrasound applications. This article explores mixed finite element formulations of classical strongly damped quasilinear models of ultrasonic wave propagation; the Kuznetsov and Westervelt equations. Such formulations allow simultaneous retrieval of the acoustic particle velocity and either the pressure or acoustic velocity potential, thus characterizing the entire ultrasonic field at once. Using non-standard energy analysis and a fixed-point technique, we establish sufficient conditions for the well-posedness, stability, and optimal a priori errors in the energy norm for the semi-discrete equations. For the Westervelt equation, we also determine the conditions under which the error bounds can be made uniform with respect to the involved strong dissipation parameter. A byproduct of this analysis is the convergence rate for the inviscid (undamped) Westervelt equation in mixed form. Additionally, we discuss convergence in the $L^q(\Omega)$ norm for the involved scalar quantities, where $q$ depends on the spatial dimension. Finally, computer experiments for the Raviart--Thomas (RT) and Brezzi--Douglas--Marini (BDM) elements are performed to confirm the theoretical findings.