Abstract

The discontinuous Galerkin (DG) time-stepping method applied to abstract evolution equation of parabolic type is studied using a variational approach. We establish the inf-sup condition or Babuška–Brezzi condition for the DG bilinear form. Then, a nearly best approximation property and a nearly symmetric error estimate are obtained as corollaries. Moreover, the optimal order error estimates under appropriate regularity assumption on the solution are derived as direct applications of the standard interpolation error estimates. Our method of analysis is new for the DG time-stepping method; it differs from previous works by which the method is formulated as the one-step method. We apply our abstract results to the finite element approximation of a second-order parabolic equation with space-time variable coefficient functions in a polyhedral domain, and derive the optimal order error estimates in several norms.

中文翻译：

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抛物方程不连续Galerkin时间步长方法的变分分析

摘要

采用变分方法研究了不连续Galerkin（DG）时间步长方法应用于抛物线型抽象演化方程。我们为DG双线性形式建立了inf-sup条件或Babuška-Brezzi条件。然后，获得近似最佳的近似性质和近似对称的误差估计作为推论。此外，在解决方案上适当规律性假设下的最佳阶数误差估计值是作为标准内插误差估计值的直接应用而得出的。我们的分析方法是DG时间步长方法的新方法。它不同于以前的方法，后者将该方法表述为一步法。我们将抽象结果应用于具有多面体域中时空可变系数函数的二阶抛物方程的有限元逼近，