Finite elements for symmetric and traceless tensors in three dimensions

TL;DR

The paper addresses the discretization of the 3D conformal deformation complex, focusing on symmetric-traceless tensors and TT fields, by constructing a conforming finite element subcomplex on tetrahedral meshes and proving exactness on contractible domains. It introduces a unified bubble–BGG framework to build trace and bubble complexes that accommodate higher vertex/surface smoothness and the linearized Cotton–York operator, enabling a stable $H(\text{div})$-conforming pairing with inf–sup stability. The main contributions include the first confirmed finite element subcomplex of the conformal complex in 3D, the establishment of divergence stability for the $H(\text{div})$-conforming pair, and the construction of $H(\cott)$-conforming spaces with appropriate DOFs and traces, yielding discrete transverse-traceless (TT) tensors and York splits in the discrete setting. The results have implications for numerical relativity and continuum mechanics, offering a structure-preserving discretization that preserves TT properties and supports York-type decompositions in computations. Overall, the work delivers a high-order, stability-assured FE framework for conformal tensor complexes, with potential extensions to distributional theories and further tensor-valued FE complexes.

Abstract

We construct a family of finite element sub-complexes of the conformal complex on tetrahedral meshes and show its exactness on contractible domains. This complex includes vector fields and symmetric and traceless tensor fields, interlinked through the conformal Killing operator, the linearized Cotton-York operator, and the divergence operator, respectively. This leads to discrete versions of transverse traceless (TT) tensors, i.e., symmetric, traceless and divergence-free matrix fields, in continuum mechanics and general relativity. We show the inf-sup stability of the $H(\operatorname{div})$-conforming finite element symmetric and traceless tensors paired with discontinuous vectors.

Figures (2)

• Figure 1: Illustration of the decomposition of a finite element space into bubbles and the rest (skeleton). At the end of the complex, $\mathop{\mathrm{div}}\nolimits$ maps bubbles onto piecewise polynomials module a finite dimensional space, which is controlled by face degrees of freedom. In general, the bubbles living on each entity (cells, faces, edges etc.) form a complex; a finite element space can be decomposed into bubbles in different dimensions plus the rest (skeleton).
• Figure 2: Vector notations on a tetrahedron $K$

Theorems & Definitions (114)

• Remark 2.1
• Remark 2.2: The interface+bubble decomposition
• Theorem 2.1: Bubble stability for the $H(\mathop{\mathrm{div}}\nolimits,\mathbb S \cap \mathbb T)\text{–}L^{2}$ pair
• Remark 2.3
• Theorem 2.2: Bubble conformal complexes
• proof : Sketch of the proof of Theorem \ref{['thm:div-surjective property']}
• Remark 2.4
• Proposition 3.1
• proof
• Proposition 3.2