论文标题
非线性声学方程的混合近似值：适当的和先验的误差分析
Mixed approximation of nonlinear acoustic equations: Well-posedness and a priori error analysis
论文作者
论文摘要
非线性声波的准确模拟对于持续开发广泛的（高强度）集中的超声应用是必不可少的。本文探讨了超声波传播的经典严重阻尼准线性模型的混合有限元公式。 Kuznetsov和Westervelt方程。这种制剂可以同时检索声学粒子速度以及压力或声速电位，从而立即表征整个超声场。使用非标准的能量分析和定点技术，我们为半分化方程的能量规范中的适当性，稳定性和最佳的先验误差建立了足够的条件。对于Westervelt方程，我们还确定了相对于涉及的强耗散参数使误差边界均匀的条件。该分析的副产品是混合形式的无粘性（未阻尼）Westervelt方程的收敛速率。此外，我们讨论了相关标量数量的$ l^q（ω）$ norm中的融合，其中$ q $取决于空间尺寸。最后，对Raviart-Thomas（RT）和Brezzi-Douglas--Marini（BDM）元素进行了计算机实验，以确认理论发现。
Accurate simulation of nonlinear acoustic waves is essential for the continued development of a wide range of (high-intensity) focused ultrasound applications. This article explores mixed finite element formulations of classical strongly damped quasilinear models of ultrasonic wave propagation; the Kuznetsov and Westervelt equations. Such formulations allow simultaneous retrieval of the acoustic particle velocity and either the pressure or acoustic velocity potential, thus characterizing the entire ultrasonic field at once. Using non-standard energy analysis and a fixed-point technique, we establish sufficient conditions for the well-posedness, stability, and optimal a priori errors in the energy norm for the semi-discrete equations. For the Westervelt equation, we also determine the conditions under which the error bounds can be made uniform with respect to the involved strong dissipation parameter. A byproduct of this analysis is the convergence rate for the inviscid (undamped) Westervelt equation in mixed form. Additionally, we discuss convergence in the $L^q(Ω)$ norm for the involved scalar quantities, where $q$ depends on the spatial dimension. Finally, computer experiments for the Raviart--Thomas (RT) and Brezzi--Douglas--Marini (BDM) elements are performed to confirm the theoretical findings.
