论文标题
直接偶然性和凸多边形的混合有限元素
Direct Serendipity and Mixed Finite Elements on Convex Polygons
论文作者
论文摘要
我们在普通刨床凸出多边形上构建了直接偶然性和直接混合有限元的新家庭，分别为$ h^1 $和$ h（div）$符合，并具有任何订单的最佳准确度。他们的自由度最少，受到一致性和准确性约束的影响。之所以出现名称，是因为形状函数是直接在物理元素上定义的，即不使用参考元素的映射。有限元形状函数定义为标量或矢量多项式的完整空间以及补充功能的空间。直接偶然性元件是De Rham复合物中直接混合元件的前体。有限元素的收敛特性在网格中多边形的形状的规律性假设下显示，以及对构建补充功能可以做出的选择的一些轻度限制。各种网格的数值实验表现出这些有限元的新家族的性能。
We construct new families of direct serendipity and direct mixed finite elements on general planer convex polygons that are $H^1$ and $H(div)$ conforming, respectively, and possess optimal order of accuracy for any order. They have a minimal number of degrees of freedom subject to the conformity and accuracy constraints. The name arises because the shape functions are defined directly on the physical elements, i.e., without using a mapping from a reference element. The finite element shape functions are defined to be the full spaces of scalar or vector polynomials plus a space of supplemental functions. The direct serendipity elements are the precursors of the direct mixed elements in a de Rham complex. The convergence properties of the finite elements are shown under a regularity assumption on the shapes of the polygons in the mesh, as well as some mild restrictions on the choices one can make in the construction of the supplemental functions. Numerical experiments on various meshes exhibit the performance of these new families of finite elements.
