A fractional-order trace-dev-div inequality

TL;DR

This paper proves a fractional-order trace-deviator-divergence inequality: for $0\le s\le 1$, $C_\mathrm{tdd}^{-1} \| \operatorname{tr} \tau \|_{H^s(\Omega)} \le \| \operatorname{dev} \tau \|_{H^s(\Omega)} + \| \operatorname{div} \tau \|_{H^{s-1}(\Omega)}$ holds for all $\tau$ in any closed subspace $\Sigma \subset H^s(\Omega;\mathbb{R}^{n\times n})$ not containing the identity, on bounded Lipschitz domains. The authors develop the necessary Sobolev interpolation framework, establish gradient/divergence mappings on $\widetilde{H}^r(\Omega)$, and prove a Poincaré-type dual bound via Bogovskii's right-inverse. They first prove the inequality for a concrete $\Sigma$ (trace-zero deviation subspace) and then extend it to general closed $\Sigma$ not containing $\mathrm{id}$ by a contradiction argument. The results yield $\lambda$-robust (Lamé parameter independent) control in mixed and least-squares finite element analyses for linear elasticity and have implications for symmetry-restricted tensor spaces.

Abstract

The trace-dev-div inequality in $H^s$ controls the trace in the norm of $H^s$ by that of the deviatoric part plus the $H^{s-1}$ norm of the divergence of a quadratic tensor field different from the constant unit matrix. This is well known for $s=0$ and established for orders $0\le s\le 1$ and arbitrary space dimension in this note. For mixed and least-squares finite element error analysis in linear elasticity, this inequality allows to establish robustness with respect to the Lamé parameter $λ$.

Theorems & Definitions (7)

• Theorem 1: tr-dev-div
• Remark 2: s=0
• Remark 3: symmetric variant
• Remark 4: s=1, n=2
• Remark 5: $\lambda$-robust regularity
• Example 6: convex domain and pure boundary condition for $n=2,3$
• Remark 7: alternative Sobolev spaces