Combined DG-CG finite element method for the Westervelt equation

TL;DR

This paper introduces a novel combined finite element method for the Westervelt equation, providing theoretical analysis and numerical validation for wave modeling in ultrasound applications.

Contribution

It develops a space-time finite element scheme with a unique energy analysis, proving well-posedness, error bounds, and asymptotic preservation for the Westervelt equation.

Findings

The scheme is well-posed and stable.

Error estimates are established.

Numerical experiments confirm theoretical results.

Abstract

We propose and analyze a space-time finite element method for Westervelt's quasilinear model of ultrasound waves in 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, \emph{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.