Year: 2025
Type: article
Abstract: We propose and analyze a space–time finite element method for Westervelt’s quasilinear model of ultrasound waves in its second-order formulation. The method combines conforming finite element spatial discretizations with a discontinuous–continuous Galerkin time stepping. Its analysis is challenged by the fact that standard Galerkin testing approaches for wave problems do not allow for bounding the discrete energy at all times. By means of redesigned energy arguments for a linearized problem combined with Banach’s fixed-point argument, we show the well-posedness of the scheme, a priori error estimates, and robustness with respect to the strong damping parameter $\delta $. Moreover, the scheme preserves the asymptotic preserving property of the continuous problem; more precisely, we prove that the discrete solutions corresponding to $\delta>0$ converge, in the singular vanishing dissipation limit, to the solution of the discrete inviscid problem. We use several numerical experiments in $(2 + 1)$ dimensions to validate our theoretical results.
Authors: Sergio Gómez, Vanja Nikolić
Language: English
Subfield: Computational Mechanics
Field: Engineering
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,718
OpenAlex ID: W4414580227