We derive an equilibrated a posteriori error estimator for the space (semi) discretization of the scalar wave equation by finite elements. In the idealized setting where time discretization is ignored and the simulation time is large, we provide fully-guaranteed upper bounds that are asymptotically constant-free and show that the proposed estimator is efficient and polynomial-degree-robust, meaning that the efficiency constant does not deteriorate as the approximation order is increased. To the best of our knowledge, this work is the first to derive provably efficient error estimates for the wave equation. We also explain, without analysis, how the estimator is adapted to cover time discretization by an explicit time integration scheme. Numerical examples illustrate the theory and suggest that it is sharp.
翻译:我们得出一个均衡的后置误差估计符, 用于用有限元素对天弧波方程式的空间( 半) 分解。 在时间分解被忽略且模拟时间大的理想环境中, 我们提供完全可靠的上界线, 且不设固定, 并显示提议的天线是高效的, 且多度- 紫色, 意指效率常数不会随着近似顺序的增加而恶化 。 据我们所知, 这项工作是第一个为波形方程式得出可辨的高效误差估计值。 我们还在不进行分析的情况下解释天线图是如何通过明确的时间集成计划来覆盖时间分解的。 数字示例说明了这个理论, 并表明它很敏锐 。