Direct serendipity and mixed finite elements on convex polygons
Direct serendipity and mixed finite elements on convex polygons
复制标题
DOI：
10.1007/s11075-022-01348-1
复制
发表时间：
2022-02
影响因子：
2.1
通讯作者：
T. Arbogast;Chuning Wang
中科院分区：
文献类型：
--
作者：
T. Arbogast;Chuning Wang
我们在一般平面严格凸多边形上分别构造了直接偶然性和直接混合有限元族，它们分别符合h （div）和h (div)，并且对任何阶都具有最优精度阶。它们具有受一致性和精度约束的最小自由度。之所以使用这个名称，是因为形状函数是直接在物理元素上定义的，也就是说，不需要使用来自引用元素的映射。有限元形状函数被定义为标量或向量多项式的满空间加上补函数的空间。直接偶然性元素是de Rham复合体中直接混合元素的前体。在网格中多边形形状的规则假设下，给出了有限元的收敛性，并对补函数的构造选择作了一些温和的限制。在不同网格上的数值实验显示了这些新的有限元族的性能。
We construct new families ofdirectserendipity anddirectmixed finite elements on general planar, strictly convex polygons that areH1andH(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 defineddirectlyon 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.