Some remarks on quadrilateral mixed finite elements
Some remarks on quadrilateral mixed finite elements
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关于四边形混合有限元的一些注记
DOI：
10.1016/j.compstruc.2008.12.006
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发表时间：
2009
影响因子：
4.7
通讯作者：
L. Gastaldi
中科院分区：
文献类型：
--
作者：
D. Boffi;L. Gastaldi
众所周知，四边形有限元用于变形网格时，可能会产生令人不满意的结果。事实证明，许多常用的有限元实现次优收敛性能的扭曲四边形;在这些元素中，我们特别回顾意外（树干）标量元素和基本上所有的向量元素的近似问题涉及的功能空间H（div）（如Raviart-Thomas或Brezzi-Douglas-Marini空间达西流）。在两个空间维度中，类似的注释适用于边缘有限元，用于近似涉及空间H（旋度）的麦克斯韦问题。另一方面，模仿有限差分已成为流行的近似问题，涉及H（div）的非常一般的几何形状。本文的目的是说明如何利用拟有限差分的思想来稳定一般四边形网格上的Raviart-Thomas单元。事实证明，这样的稳定可以通过对标准Raviart-Thomas元素的轻微修改来执行，这不会显着增加原始方案的计算成本。
It is well known that quadrilateral finite elements may produce unsatisfactory results when used on distorted meshes. It turns out that many commonly used finite elements achieve suboptimal convergence properties on distorted quadrilaterals; among such elements we recall in particular serendipity (trunk) scalar elements and basically all vectorial elements for the approximation of problems involving the functional space H(div) (like Raviart–Thomas or Brezzi–Douglas–Marini spaces for Darcy flow). In two space dimensions, a similar remark applies to edge finite elements for the approximation of Maxwell’s problems involving the space H(curl). On the other hand, mimetic finite differences have become popular for the approximation of problems involving H(div) on very general geometries. The aim of this paper is to show how to use the ideas of mimetic finite differences for the stabilization of Raviart–Thomas element on general quadrilateral meshes. It turns out that such stabilization can be performed by a slight modification of the standard Raviart–Thomas element which does not increase significantly the computational cost of the original scheme.