Submitted

Finite Element Complexes in Two Dimensions

Long Chen and Xuehai Huang

Submitted

arXiv   Bibtex

ABSTRACT:

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.