Submitted

Basis Construction for Smooth Finite Element Spaces

Chunyu Chen, Long Chen, Tingyi Gao, Xuehai Huang, and

Submitted

arXiv   Bibtex

ABSTRACT:

We give a general implementable construction of local and
global bases for the $C^m$-conforming polynomial finite element spaces
of Hu, Lin, and Wu on simplicial meshes. Although these spaces and
their unisolvent degrees of freedom are known, explicit basis
functions suitable for computation were not previously available in
arbitrary dimension and smoothness order. Using the geometric
decomposition of the simplicial lattice developed by Chen and Huang,
we replace the original moment functionals by Bernstein-dual
functionals on subsimplices. Together with a dual normal frame, this
choice yields a lower-triangular local DoF--basis matrix whose
diagonal blocks are positive scalar multiples of the identity. The
local basis is therefore computed by triangular substitution. We
further introduce global normal frames and derive an explicit
local-to-global basis transformation for conforming assembly. In
addition, we obtain recursive dimension formulas for these smooth
finite element spaces and recover the standard $C^0$ Lagrange
dimension formula as a special case. The construction is implemented
in FEALPy. Numerical tests for interpolation and polyharmonic
equations in two and three dimensions confirm the expected
approximation orders.