A priori error estimate for the approximate $C^1$ isogeometric discretization

Let $\varphi_h\in\widetilde{\mathcal{V}}^1_{h,0}$ solve the discrete problem on the two-patch domain, and let $p_1^{(S)}\geq 2$, $p_2\geq 3$, and $r_1,r_2,\widetilde{r}\geq 1$. Define

$$q=\min\bigl(p_1^{(L)}-1,p_1^{(R)}-1,p_2-1,\widetilde{p}+1\bigr)$$

if $\widetilde{\alpha}^{(S)},\widetilde{\beta}^{(S)}\notin\mathbb{P}^{\widetilde{p}}$, and otherwise define

$$q=\min\bigl(p_1^{(L)}-1,p_1^{(R)}-1,p_2-1\bigr).$$

Suppose that $\varphi$ solves the model problem, $\varphi\in H^{5/2+\varepsilon}(\Omega)$, and $\varphi|_{\Omega^{(S)}}\in H^{q+2}(\Omega^{(S)})$. A priori error estimate. Under the assumptions of Theorem~,

$$\left\lVert\varphi-\varphi_h\right\rVert_{\mathcal{X}}\leq Ch^q\sum_{S\in\{L,R\}}\left\lVert\varphi\right\rVert_{H^{q+2}(\Omega^{(S)})}.$$

This estimate predicts convergence of order $q$ in the $\mathcal{X}$-norm for the approximate $C^1$ discretization; the rate is limited by the spline degrees and by the approximation of the gluing data. The source presents it as an expected estimate, and no resolution status is supplied.