论文标题
一般四边形电网上的最低度分段多项式DE RHAM复合物
Lowest-degree piecewise polynomial de Rham complex on general quadrilateral grids
论文作者
论文摘要
本文致力于在不限于平行四边形的一般四边形组成的网格上构建有限元。分别针对$ h^1 $和$ h（\ rm ot）$椭圆问题建立了两个有限元素。 $ \ MATHCAL {O}（H）$ contergence and of and of and and $ \ Mathcal {O}（H^2）$ l^2 $ norm in $ h^1 $方案的$ l^2 $ norm in Anryptotic-Parallex-Parallex假设在网格上证明了。此外，一般四边形网格上的两个有限元空间，以及分段恒定功能的空间，制定了离散的DE RHAM复合物。 有限的元件空间由分段多项式函数组成，因此在一般四边形网格上不合格。实际上，本文对严格的分析进行了严格的分析，即无法构建一个实际有用的有限元素，该元素定义为Ciarlet的三重元素，该元件可以制定有限元元素空间，该元件由连续的分段多项式函数组成，该函数可能包括任意四边形。
This paper is devoted to the construction of finite elements on grids that consist of general quadrilaterals not limited in parallelograms. Two finite elements defined as Ciarlet's triple are established for the $H^1$ and $H(\rm rot)$ elliptic problems, respectively. An $\mathcal{O}(h)$ order convergence rate in energy norm for both of them and an $\mathcal{O}(h^2)$ order convergence in $L^2$ norm for the $H^1$ scheme are proved under the asymptotic-parallelogram assumption on the grids. Further, the two finite element spaces on general quadrilateral grids, together with the space of piecewise constant functions, formulate a discretized de Rham complex. The finite element spaces consist of piecewise polynomial functions, and, thus, are nonconforming on general quadrilateral grids. Indeed, a rigorous analysis is given in this paper that it is impossible to construct a practically useful finite element defined as Ciarlet's triple that can formulate a finite element space which consists of continuous piecewise polynomial functions on a grid that may include arbitrary quadrilaterals.
