Poincaré型不等式在多边形域中常数的精确界
Sharp bounds of constants in Poincare type inequalities for polygonal domains
AI总结：
本文研究了在Lipschitz域边界上具有零均值函数的Poincaré型不等式常数的精确估计，针对数值分析中典型的简单形域，提出易于计算的精确界，并通过数值分析比较了经典Poincaré不等式常数与已知解析估计。
摘要：
本文研究了在Lipschitz域边界或其可测部分上具有零均值函数的Poincaré型不等式常数的精确估计。这些估计对各种数值方法，特别是偏微分方程（PDEs）的后验误差估计方法非常有用。因此，我们主要关注数值分析中典型的简单形域（二维和三维），并提出易于计算的关系式，以提供相应的精确界。此外，我们通过数值分析研究了经典Poincaré不等式中常数的行为，并将其结果与已知的解析估计进行比较。在最后一节中，这些估计被用于获得椭圆边值问题的新后验估计。
英文摘要：
The paper is concerned with sharp estimates of constants in Poincare type inequalities for functions having zero mean value on the boundary of a Lipschitz domain or on a measurable part of it. These estimates are useful for various numerical methods, in particular, for a posteriori error estimation methods for partial differential equations (PDEs). Therefore, we are mainly focused on domains typical for numerical analysis (simplexes in 2d and 3d) and suggest easily computable relations that provide sharp bounds of the respective constants. Also, we investigate numerically the behavior of the constants in the classical Poincare inequalities and compare these results with known analytical estimates. In the last section, the estimates are used in order to obtain new a posteriori estimates for an elliptic boundary value problem.
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