论文标题
有限元络合物二维
Finite Element Complexes in Two Dimensions
论文作者
论文摘要
在这项研究中，系统地构建了具有不同水平平滑度的二维有限元复合物，包括DE RHAM复合物，Curldiv复合物，弹性复合物和Divdiv复合物。基于简单晶格的非重叠分解和多项式空间的伯恩斯坦基础，开发了两个维度中的平滑标量有限元，而在顶点的可怜性顺序大于边缘的两倍。使用具有平滑度参数的平滑有限元素来设计具有不同级别平滑度的有限元元素DE RHAM复合物，可以满足某些关系。最后，使用Bernstein-Gelfand-Gelfand（BGG）框架，有限元弹性复合物和有限元divdiv复合物是从有限元元素DE RHAM复合物中得出的。这项研究是第一个以系统的方式构建有限元复合物的工作。此外，这项工作中开发的新工具，例如简单晶格的非重叠分解和离散的BGG结构，可用于该领域的进一步研究。
In this study, two-dimensional finite element complexes with various levels of smoothness, including the de Rham complex, the curldiv complex, the elasticity complex, and the divdiv complex, are systematically constructed. 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, with the order of differentiability at vertices being greater than twice that at edges. Finite element de Rham complexes with different levels of smoothness are devised using smooth finite elements with smoothness parameters that satisfy 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. This study is the first work to construct finite element complexes in a systematic way. Moreover, the novel tools developed in this work, such as the non-overlapping decomposition of the simplicial lattice and the discrete BGG construction, can be useful for further research in this field.
