Finite Element Complexes with Traces Structures: A unified framework for cohomology and bounded interpolation

2025/09/28 by Hu, Jun, Liang, Yizhou, Lin, Ting
#FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.2509.23788

Abstract

This paper considers the cohomology and bounded interpolation of nonstandard finite element complexes, e.g. Stokes, Hessian, Elasticity, divdiv. Compared to the standard finite element exterior calculus, the main challenge is the existence of extra smoothness. This paper provides a unified framework for finite element complexes with extra smoothness. The trace structure is introduced to derive the bubble complexes in different dimensions (vertices, edges, faces). It is shown that if the bubble complexes in different dimensions are all exact, then the finite element has the correct cohomology. Moreover, the L² bounded interpolation can be constructed.

Citations

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Related

• Finite Element Complexes in Two Dimensions
• Complexes from Complexes: Finite Element Complexes in Three Dimensions
• Finite Element de Rham and Stokes Complexes in Three Dimensions
• Nonstandard finite element de Rham complexes on cubical meshes
• Higher-order finite element de Rham complexes, partially localized flux reconstructions, and applications