Comparison Results of Finite Element Methods for the Poisson Model Problem
Comparison Results of Finite Element Methods for the Poisson Model Problem
复制标题
泊松模型问题的有限元方法比较结果
DOI：
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发表时间：
2012
影响因子：
2.9
通讯作者：
M. Schedensack
中科院分区：
文献类型：
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作者：
C. Carstensen;D. Peterseim;M. Schedensack
本文在Poisson模型问题的能量误差范数与一般常数和高阶数据振荡相等的意义上，建立了合态Courant有限元法、非合态Crouzeix—Raviart有限元法和几种一阶不连续Galerkin有限元法的等价性。Raviart—Thomas混合有限元法优于以往的方法，但证明了逆关系的猜想是错误的。本文完成了Braess发起的比较分析[Calcolo， 46 (2009), pp. 149—155]。两个数值基准说明了比较定理和Raviart- Thomas混合有限元法可能的严格优越性。应用包括最小二乘有限元方法，有限体积方法，以及自适应有限元方法中最优性概念的近似类的等式。
This paper establishes the equivalence of the conforming Courant finite element method, the nonconforming Crouzeix--Raviart finite element method, and several first-order discontinuous Galerkin finite element methods in the sense that the respective energy error norms are equivalent up to generic constants and higher-order data oscillations in a Poisson model problem. The Raviart--Thomas mixed finite element method is better than the previous methods, whereas the conjecture of the converse relation is proved to be false. This paper completes the analysis of comparison initiated by Braess [Calcolo, 46 (2009), pp. 149--155]. Two numerical benchmarks illustrate the comparison theorems and the possible strict superiority of the Raviart--Thomas mixed finite element method. Applications include least-squares finite element methods, finite volume methods, and equality of approximation classes for concepts of optimality for adaptive finite element methods.