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de Rham复形、旋度散度复形、弹性复形及散度散度复形。基于单纯形格子的非重叠分解与多项式空间的Bernstein基，发展了二维光滑标量有限元，其在顶点处的可微阶数需达到边处阶数的两倍以上。通过构造满足特定关系的光滑参数有限元，设计了具有不同光滑度水平的有限元de Rham复形。最终，利用Bernstein-Gelfand-Gelfand（BGG）框架从有限元de Rham复形导出了有限元弹性复形与有限元散度散度复形。本研究首次以系统性方式构造有限元复形，此外，本文发展的创新工具（如单纯形格子的非重叠分解及离散BGG构造）可为该领域的后续研究提供重要参考。