This paper presents a general framework for the coercivity analysis of a class of quadratic finite volume element (FVE) schemes on triangular meshes for solving elliptic boundary value problems. This class of schemes covers all the existing quadratic schemes of Lagrange type. With the help of a new mapping from the trial function space to the test function space, we find that each element matrix can be decomposed into three parts: the first part is the element stiffness matrix of the standard quadratic finite element method (FEM), the second part is the difference between the FVE and FEM on the element boundary, while the third part can be expressed as the tensor product of two vectors. Thanks to this decomposition, we obtain a sufficient condition to guarantee the existence, uniqueness, and coercivity result of the FVE solution on triangular meshes. Moreover, based on this sufficient condition, some minimum angle conditions with simple, analytic, and computable expressions can be derived and they depend only on the constructive parameters of the schemes. As a byproduct, some existing coercivity results are improved. Finally, an optimal H¹ error estimate is proved by the standard techniques.

中文翻译：

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三角网格上一类二次有限体积单元格式的统一分析

本文为解决椭圆形边值问题的三角网格上的一类二次有限体积单元（FVE）方案的矫顽性分析提供了一个通用框架。此类方案涵盖了Lagrange类型的所有现有二次方案。借助于从试验函数空间到测试函数空间的新映射，我们发现每个元素矩阵都可以分解为三个部分：第一部分是标准二次有限元方法（FEM）的元素刚度矩阵，第二部分是元素边界上的FVE和FEM之差，而第三部分可以表示为两个向量的张量积。通过这种分解，我们获得了足够的条件来保证存在，唯一性，三角形网格上的FVE解的矫顽力结果。此外，基于此充分条件，可以推导出一些具有简单，解析和可计算表达式的最小角度条件，它们仅取决于方案的构造参数。作为副产品，一些现有的矫顽力结果得到了改善。最后，一个最优H ¹误差估计由标准技术证明。