We derive a posteriori error estimates for the the scalar wave equation discretized in space by continuous finite elements and in time by the explicit leapfrog scheme. Our analysis combines the idea of invoking extra time-regularity for the right-hand side, as previously introduced in the space semi-discrete setting, with a novel, piecewise quartic, globally twice-differentiable time-reconstruction of the fully discrete solution. Our main results show that the proposed estimator is reliable and efficient in a damped energy norm. These properties are illustrated in a series of numerical examples.

翻译：我们针对空间上采用连续有限元离散、时间上采用显式蛙跳格式离散的标量波动方程，推导了后验误差估计。本文分析结合了之前在空间半离散框架中引入的通过右端项额外时间正则性思想，并引入了一种新的、分段四次、全局二阶可微的全离散解时间重构方法。主要结果表明，所提出的估计器在阻尼能量范数下具有可靠性和有效性。通过一系列数值算例验证了这些性质。