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

Abstract

Accurate simulation of nonlinear acoustic waves is essential for the continued development of a wide range of (high-intensity) focused ultrasound applications. In this article, we explore 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 complete 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 dissipation parameter. Additionally, we discuss convergence in the maximum error norm of the involved scalar quantities. Finally, computer experiments for Raviart–Thomas (RT) and Brezzi–Douglas–Marini (BDM) elements are performed to confirm the theoretical findings.