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，本文研究了Westervelt型拟线性（可能非局部）方程的鲁棒协调有限元离散格式。这些波动方程涉及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和正性假设的记忆核，将其统一归为一类。由于所涉及的阻尼参数相对较小，且在某些无粘性介质中可忽略不计，因此开发在参数消失时仍保持稳定的方法至关重要。为此，本文工作具有双重贡献：首先，我们确定了使（非）局部Westervelt方程的协调有限元离散对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设计适当的能量泛函，使其相对于阻尼参数保持一致有界。