  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.
