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.

翻译：本研究系统构造了具有不同光滑度水平的二维有限元复形，包括德拉姆复形、旋度散度复形、弹性复形和散度-散度复形。基于单纯格的非重叠分解和多项式空间的伯恩斯坦基，发展了二维光滑标量有限元，其顶点处的可微性阶数大于边处可微性阶数的两倍。利用满足特定关系的光滑参数的光滑有限元，设计了具有不同光滑度水平的有限元德拉姆复形。最后，通过伯恩斯坦-盖尔范德-盖尔范德（BGG）框架，从有限元德拉姆复形推导出有限元弹性复形和有限元散度-散度复形。本研究是首次以系统化方式构造有限元复形的工作。此外，本研究中开发的新工具，如单纯格的非重叠分解和离散BGG构造，可为本领域的进一步研究提供有益参考。