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October 20, 2025Open Access

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

Key Points

The correct cohomology is ensured when all bubble complexes in various dimensions are exact.
Introducing trace structures facilitates deriving bubble complexes across dimensions, such as vertices and edges.
Extra smoothness presents challenges compared to standard finite element exterior calculus.
The paper constructs $L^2$ bounded interpolation as part of the unified framework.

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.

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

Key Points

Abstract

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