Submitted

Implementation and Basis Construction for Smooth Finite Element Spaces

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

Submitted

arXiv   Bibtex

ABSTRACT:

The construction of $C^m$ conforming finite elements on
simplicial meshes has recently advanced through the groundbreaking
work of Hu, Lin, and Wu (Found. Comput. Math. 24, 2024). Their
framework characterizes smoothness via moments of normal derivatives
over subsimplices, leading to explicit degrees of freedom and
unisolvence, unifying earlier constructions. However, the absence of
explicit basis functions has left these spaces largely inaccessible
for practical computation. In parallel, multivariate spline theory
(Chui and Lai, J. Approx. Theory 60, 1990) enforces $C^m$ smoothness
through linear constraints on Bernstein--B\'{e}zier coefficients, but
stable, locally supported bases remain elusive beyond low
dimensions. Building on the geometric decomposition of the simplicial
lattice proposed by Chen and Huang (Math. Comp. 93, 2024), this work
develops an explicit, computable framework for smooth finite
elements. The degrees of freedom defined by moments of normal
derivatives are modified to align with the dual basis of the Bernstein
polynomials, yielding structured local bases on each simplex. Explicit
basis construction is essential not merely for completeness, but for
enabling efficient matrix assembly, global continuity, and scalable
solution of high-order elliptic partial differential equations. This
development closes the gap between theoretical existence and practical
realization, opening smooth finite element methods to broad
computational application.