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, 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.

翻译：本文针对二阶形式的Westervelt超声波动拟线性模型，提出并分析了一种时空有限元方法。该方法将协调有限元空间离散化与间断-连续Galerkin时间步进相结合。其分析面临以下挑战：波动问题的标准Galerkin测试方法无法始终保证离散能量的有界性。通过为线性化问题重构能量论证并结合Banach不动点定理，我们证明了该格式的适定性、先验误差估计以及对强阻尼参数$\delta$的鲁棒性。此外，该格式保持了连续问题的渐近保持特性；更精确地说，我们证明了对应于$\delta>0$的离散解在奇异消失耗散极限下收敛于离散无粘性问题的解。我们通过多个$(2+1)$维数值实验验证了理论结果。