标题： 基于间断伽辽金-连续伽辽金有限元方法的Westervelt方程 显示英文标题

标题： Combined DG-CG finite element method for the Westervelt equation

摘要： 我们提出并分析了一种时空有限元方法，用于Westervelt提出的声波二阶形式的准线性模型。 该方法结合了保形有限元空间离散化与间断-连续Galerkin时间步进法。 由于标准波动问题的Galerkin测试方法无法在所有时刻控制离散能量这一事实，其分析面临挑战。 通过重新设计的能量论证方法（针对线性化问题）以及Banach不动点论证，我们证明了该方案的适定性、\emph{先验地}误差估计以及对强阻尼参数$\delta$的鲁棒性。 此外，该方案保留了连续问题的渐近保持性质；更确切地说，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到离散无粘问题的解。 我们使用多个二维$(2 + 1)$维度的数值实验验证了我们的理论结果。 显示英文摘要

摘要： 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.