Finite-Lamé isogeometric elasticity is conforming and quasi-optimal

Given: Countable Choice; the elasticity data and domain above; and a finite isogeometric space declared conforming in $H_0^1(D;{\mathbb{R}}^d)$.

Conforming isogeometric Galerkin spaces and h/p/k refinement: the scalar local space is finite dimensional; the elasticity space is its componentwise $d$-fold product restricted to global $H_0^1(D;{\mathbb{R}}^d)$; nonmatching faces are excluded.

Matching NURBS patch traces give global H1 conformity: under Countable Choice, on a bounded open set in dimension two or three with a specified finite face-to-face piecewise $C^1$ partition into bounded knot-cell images with disjoint interiors, a field $C^1$ on each closed cell, with square-integrable cell gradients and equal one-sided values on every shared face, belongs to global $H^1$.

Linear isotropic elasticity: displacement form and data fixes the real convention $V=H_0^1(D;{\mathbb{R}}^d)$ with gradient norm, $\mu \in L^{\mathrm{\infty }}$ with $0<{\mu }_0\le \mu \le {\mu }_1$, $\lambda \ge 0$, $f\in V^{\mathrm{\prime }}$, $\epsilon (v)=(\mathrm{\nabla }v+\mathrm{\nabla }{v}^T)\mathrm{/}2$, and ${\mathcal{E}}_{\lambda }(v,w)=2{\int }_D\mu \epsilon (v):\epsilon (w)+\lambda {\int }_D\mathrm{div}v\mathrm{div}w$.

Dirichlet elasticity energy bound and Galerkin estimate states: under Countable Choice, $d\in \{2,3\}$, the data of [F3], finite $\lambda \ge 0$, and any finite-dimensional conforming $V_h\subset V$, the continuous and Galerkin solutions are unique and $\mathrm{\parallel }u-u_h{\mathrm{\parallel }}_V\le (2{\mu }_1+d\lambda ){\mu }_0^{-1}{\inf }_{v_h\in V_h}\mathrm{\parallel }u-v_h{\mathrm{\parallel }}_V$.

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$) is the exact axiom required by the imported elasticity theorem and its Hilbert, Lax--Milgram and Céa applications; this item adds no stronger choice principle.