In this paper, we analyze the discrete inf-sup condition and related error estimates for a modified Hilbert transformation as used in the space-time discretization of time-dependent partial differential equations. It turns out that the stability constant depends linearly on the finite element mesh size . While the ratio decreases as for , numerical results indicate a decay of for some in the polynomial degree of the finite element basis functions. However, in most cases, we can show optimal convergence. We present a series of numerical experiments which illustrate the theoretical findings.

中文翻译：

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关于改进的希尔伯特变换、离散 inf-sup 条件和误差估计


在本文中，我们分析了时相关偏微分方程的时空离散化中使用的修正希尔伯特变换的离散 inf-sup 条件和相关误差估计。事实证明，稳定常数与有限元网格尺寸线性相关。虽然 的比率减小，但数值结果表明有限元基函数的多项式次数中的某些值有所衰减。然而，在大多数情况下，我们可以表现出最佳的收敛性。我们提出了一系列数值实验来说明理论结果。