We consider the numerical approximation of acoustic wave propagation in the time domain by a mixed finite element method based on the BDM1-P0 spaces. A mass-lumping strategy for the BDM1 element, originally proposed by Wheeler and Yotov in the context of subsurface flow problems, is utilized to enable an efficient integration in time. By this mass-lumping strategy, the accuracy of the mixed method is formally reduced to first order. We will show, however, that the numerical approximation still carries global second order information, which is expressed as super-convergence of the numerical approximation to certain projections of the true solution. Based on this fact, we propose post-processing strategies for both variables, the pressure and the velocity, which yield piecewise linear approximations of second order accuracy. A complete convergence analysis is provided for the semi-discrete and corresponding fully-discrete approximations, which result from time discretization by the leapfrog method. In addition, some numerical tests are presented to illustrate the efficiency of the proposed approach.

中文翻译：

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一种用于声波传播的质量集总混合有限元方法

我们通过基于 BDM1-P0 空间的混合有限元方法来考虑时域中声波传播的数值近似。最初由 Wheeler 和 Yotov 在地下流动问题的背景下提出的 BDM1 元素的质量集总策略用于实现有效的时间整合。通过这种质量集总策略，混合方法的精度正式降低到一阶。然而，我们将证明数值近似仍然携带全局二阶信息，这表示为数值近似对真解的某些投影的超收敛。基于这一事实，我们提出了压力和速度这两个变量的后处理策略，这会产生二阶精度的分段线性近似。对半离散和相应的全离散近似提供了完整的收敛分析，这些近似是由跳跃法的时间离散化产生的。此外，还提供了一些数值测试来说明所提出方法的效率。