抛物问题中 L∞(0,T;L2(Ω)) 的对偶与能量后验估计比较
A comparison of duality and energy aposteriori estimates for L?(0,T;L2(Ω)) in parabolic problems
AI总结：
本文利用椭圆重构技术和对偶方法，针对线性抛物方程的全离散向后欧拉格式推导后验误差估计，并与能量方法比较，得到L∞(0,T;L2(Ω))范数下的最优阶估计。
摘要：
我们使用椭圆重构技术结合对偶方法，为线性抛物方程的全离散向后欧拉格式证明后验误差估计。作为应用，我们将我们的结果与椭圆问题后验估计中的残差基估计器相结合，推导空间误差指示器，从而得到在 L∞(0,T;L2(Ω)) 范数下误差界的完全实用版本。这些最优阶估计器通过考虑网格变化引起的误差并允许更灵活地使用椭圆估计器，扩展了 Eriksson 和 Johnson (1991) 引入的估计器。为了与先前结果比较，我们还推导了基于能量的 L∞(0,T;L2(Ω)) 误差后验估计，它简化了 Lakkis 和 Makridakis (2006) 中给出的一个估计。然后我们在实际情况下比较这两种估计器（对偶与能量）并得出结论。
英文摘要：
We use the elliptic reconstruction technique in combination with a duality approach to prove aposteriori error estimates for fully discrete back- ward Euler scheme for linear parabolic equations. As an application, we com- bine our result with the residual based estimators from the aposteriori esti- mation for elliptic problems to derive space-error indicators and thus a fully practical version of the estimators bounding the error in the L \infty (0, T ; L2(Ω)) norm. These estimators, which are of optimal order, extend those introduced by Eriksson and Johnson (1991) by taking into account the error induced by the mesh changes and allowing for a more flexible use of the elliptic estima- tors. For comparison with previous results we derive also an energy-based aposteriori estimate for the L \infty (0, T ; L2(Ω))-error which simplifies a previous one given in Lakkis and Makridakis (2006). We then compare both estimators (duality vs. energy) in practical situations and draw conclusions.
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