Two-dimensional finite element complexes with various smoothness, including the de Rham complex, the curldiv complex, the elasticity complex, and the divdiv complex, are systematically constructed in this work. First smooth scalar finite elements in two dimensions are developed based on a non-overlapping decomposition of the simplicial lattice and the Bernstein basis of the polynomial space. Smoothness at vertices is more than doubled than that at edges. Then the finite element de Rham complexes with various smoothness are devised using smooth finite elements with smoothness parameters satisfying certain relations. Finally, finite element elasticity complexes and finite element divdiv complexes are derived from finite element de Rham complexes by using the Bernstein-Gelfand-Gelfand (BGG) framework. Additionally, some finite element divdiv complexes are constructed without BGG framework. Dimension count plays an important role for verifying the exactness of two-dimensional finite element complexes.
翻译：在这项工作中,系统构建了包括德拉姆综合体、卷轴综合体、弹性综合体和迪夫迪夫综合体等各种平稳的二维有限元素综合体。在不重叠地分解复合空间的简化板状和伯尔尼斯坦基基的基础上,开发了两个层面的首个光滑的伸缩元素。顶部的滑动比边缘的平滑多一倍多。然后,利用光滑参数的平滑有限元素设计出具有各种关系的拉姆综合体。最后,利用伯恩斯坦-盖尔凡-盖尔芬德(BGG)框架,从有限元素中衍生出有限弹性复合体的缩缩缩放复合体和有限要素。此外,在没有BGGG框架的情况下,建造了一些有限元素divdiv综合体。尺寸计数在核实二维综合体的准确性方面起着重要作用。
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