Motivated by numerical modeling of ultrasound waves, we investigate robust conforming finite element discretizations of quasilinear and possibly nonlocal equations of Westervelt type. These wave equations involve either a strong dissipation or damping of fractional-derivative type and we unify them into one class by introducing a memory kernel that satisfies non-restrictive regularity and positivity assumptions. As the involved damping parameter is relatively small and can become negligible in certain (inviscid) media, it is important to develop methods that remain stable as the said parameter vanishes. To this end, the contributions of this work are twofold. First, we determine sufficient conditions under which conforming finite element discretizations of (non)local Westervelt equations can be made robust with respect to the dissipation parameter. Secondly, we establish the rate of convergence of the semi-discrete solutions in the singular vanishing dissipation limit. The analysis hinges upon devising appropriate energy functionals for the semi-discrete solutions that remain uniformly bounded with respect to the damping parameter.

翻译：受超声波数值模拟的驱动，我们研究了Westervelt型拟线性及可能非局部方程的鲁棒协调有限元离散化方法。这类波动方程包含强耗散或分数阶导数型阻尼，我们通过引入满足非限制性正则性与正定性假设的记忆核将其统一为一类方程。由于所涉阻尼参数相对较小，在某些（无粘性）介质中可忽略不计，因此开发在该参数趋近于零时仍保持稳定的方法至关重要。为此，本工作的贡献体现在两个方面：首先，我们确定了使（非）局部Westervelt方程的协调有限元离散化关于耗散参数具有鲁棒性的充分条件；其次，我们建立了半离散解在奇异消失耗散极限下的收敛速率。分析的关键在于为半离散解构造适当的能量泛函，该泛函关于阻尼参数保持一致有界。