Implementation and Basis Construction for Smooth Finite Element Spaces

TL;DR

This paper develops an explicit, computable basis for smooth finite element spaces on simplicial meshes, bridging the gap between theoretical constructions and practical computational methods for high-order PDEs.

Contribution

It introduces a structured local basis aligned with the dual Bernstein basis, enabling efficient implementation of smooth finite element methods.

Findings

Explicit basis functions are constructed for $C^m$ finite element spaces.

The basis facilitates efficient matrix assembly and scalable PDE solutions.

Bridges the gap between theoretical smoothness conditions and practical computation.

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, making smooth finite element methods accessible to broad computational applications.