广义Beta素分布在有限元误差逼近中的应用
Generalized Beta Prime Distribution Applied to Finite Element Error Approximation
摘要：
本文提出了一种基于广义Beta素数分布的新一代概率律，用以估计两种Lagrange有限元$P_{k_1}$和$P_{k_2}, (k_1<k_2)$之间的相对精度。由于相对有限元精度通常基于网格步长$h$变为零时的渐近收敛速度的比较，因此该概率定律强调存在$P_{k_1}$有限元比$P_{k_2}$更可能精确的情况，这取决于$h$。为了证实这一特点，我们显示和检查的实际例子，统计频率和相应的概率确定的概率定律之间的拟合质量。其中，它验证了当$h$远离零时，有限元$P_{k_1}$可以产生比有限元$P_{k_2}$更精确的结果，因为事件“$P_{k_1}$比$P_{k_2}$更精确“的概率因此增加到大于0.5。在这些情况下，$P_{k_2}$有限元更有可能被过度限定。
英文摘要：
In this paper we propose a new generation of probability laws based on the generalized Beta prime distribution to estimate the relative accuracy between two Lagrange finite elements $P_{k_1}$ and $P_{k_2}, (k_1<k_2)$. Since the relative finite element accuracy is usually based on the comparison of the asymptotic speed of convergence when the mesh size $h$ goes to zero, this probability laws highlight that there exists, depending on $h$, cases such that $P_{k_1}$ finite element is more likely accurate than the $P_{k_2}$ one. To confirm this feature, we show and examine on practical examples, the quality of the fit between the statistical frequencies and the corresponding probabilities determined by the probability law. Among others, it validates, when $h$ moves away from zero, that finite element $P_{k_1}$ may produces more precise results than a finite element $P_{k_2}$ since the probability of the event "$P_{k_1}$ is more accurate than $P_{k_2}$" consequently increases to become greater than 0.5. In these cases, $P_{k_2}$ finite elements are more likely overqualified.
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