Abstract The aim of this paper is to provide new perspectives on the relative finite elements accuracy. Starting from a geometrical interpretation of the error estimate which can be deduced from Bramble–Hilbert lemma, we derive a probability law that evaluates the relative accuracy, considered as a random variable, between two finite elements Pk and Pm, k < m. We extend this probability law to get a cumulated probabilistic law for two main applications. The first one concerns a family of meshes, the second one is dedicated to a sequence of simplexes constituting a given mesh. Both of these applications could be considered as a first step toward application for adaptive mesh refinement with probabilistic methods.

中文翻译：

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关于评估有限元精度的广义二项式定律：自适应网格细化的初步概率结果

摘要 本文的目的是为相对有限元精度提供新的视角。从可以从 Bramble-Hilbert 引理推导出的误差估计的几何解释开始，我们推导出一个概率定律，该概率定律评估两个有限元素 Pk 和 Pm 之间的相对精度，被视为随机变量，k < m。我们扩展这个概率定律以获得两个主要应用的累积概率定律。第一个涉及网格系列，第二个专门用于构成给定网格的一系列单纯形。这两种应用都可以被视为应用概率方法自适应网格细化应用的第一步。