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NURBS quotient spaces transfer numerator H1 approximation

Statement

Assume Countable Choice. Let $F : \hat{Ω} \to Ω$ be an admissible NURBS map and $W$ its continuous denominator, with $W \geq w_{-} > 0$ and cellwise $G_{W} : = ∥ \nabla W ∥_{L^{\infty}} < \infty$ . Let $\hat{u}$ be real-valued and let $s \in S_{h}$ be a real numerator spline; assume both are $C^{1}$ up to the boundary of each cell in the common refinement of the geometry and analysis knot grids, with matching traces, and put $g : = W \hat{u}$ . Then the physical rational spline $v_{h} : = (s / W) \circ F^{- 1}$ belongs to $H^{1} (Ω; R)$ and $∥ u - v_{h} ∥_{H^{1} (Ω)} \leq C_{F} C_{W} ∥ g - s ∥_{H^{1} (\hat{Ω})}, C_{F} : = \sqrt{j_{+}} \sqrt{1 + L^{2}}, C_{W} : = \frac{2}{w_{-}} + \frac{G_{W}}{w_{-}^{2}},$ where $u : = \hat{u} \circ F^{- 1}$ . This result transfers a specified numerator approximation bound; it does not assert a polynomial-degree-independent rate or a general $H^{1}$ rate for arbitrary knot families.

Facts & Assumptions

Given: Countable Choice; the admissible patch map $F$ , continuous positive denominator $W$ , real-valued parameter function $\hat{u}$ , real numerator spline $s$ , and matching cell traces of the Statement.

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ) is the exact choice assumption of the mapped H1 supplier [F2]. The local finite-cell quotient estimate requires no stronger axiom.

[F1]

Positive weights give a bounded partition-of-unity NURBS basis: $W \geq w_{-} > 0$ , $\nabla W$ is essentially bounded on the finite cells, and $N_{h} = S_{h} / W$ .

[F2]

Bi-Lipschitz NURBS maps preserve H1 and energy norms: $H^{1}$ norms on $\hat{Ω}$ and $Ω$ are equivalent with the stated map constants.

[F3]

Matching NURBS patch traces give global H1 conformity: a piecewise $C^{1}$ function with matching traces and square-integrable cell gradients belongs to global $H^{1}$ .

Proof

1.1F1F3givenalgebra

Work on the finite common refinement and set $e : = g - s$ . The continuous positive denominator is polynomial on each geometry cell, so $g$ , $s$ , $e$ , $s / W$ and $e / W$ are continuous and C1 up to each refined cell boundary. Their classical cell gradients are square integrable on this finite partition; [F3] therefore puts them in H1 with these weak gradients. On each cell, $u - v_{h} = (\hat{u} - s / W) \circ F^{- 1} = (e / W) \circ F^{- 1}$ . Since $W \geq w_{-} > 0$ , the cellwise quotient rule gives $∣ \frac{e}{W} ∣ \leq \frac{∣ e ∣}{w_{-}}, ∣ \nabla ⁣ (\frac{e}{W}) ∣ \leq \frac{∣ \nabla e ∣}{w_{-}} + \frac{G_{W} ∣ e ∣}{w_{-}^{2}} .$ Taking the $H^{1}$ norm and using the triangle inequality yields $∥ e / W ∥_{H^{1} (\hat{Ω})} \leq C_{W} ∥ e ∥_{H^{1} (\hat{Ω})}$ .

2.1A1F2F3step 1.1algebra∎

The traces of $e / W$ match because the traces of $\hat{u}$ , $s$ , and $W$ match and $W$ is positive. Its cell gradients are square integrable by step 1.1, so [F3] gives $e / W \in H^{1} (\hat{Ω})$ . By [A1], [F2] maps $s / W \in H^{1} (\hat{Ω})$ to $v_{h} \in H^{1} (Ω)$ . Applying its inverse-map estimate to $(e / W) \circ F^{- 1}$ gives $∥ u - v_{h} ∥_{H^{1} (Ω)} \leq C_{F} ∥ e / W ∥_{H^{1} (\hat{Ω})} \leq C_{F} C_{W} ∥ g - s ∥_{H^{1} (\hat{Ω})} .$ This is the claimed estimate, with all constants retaining the map and weight dependence.

Depends on

Used by

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Sources

Stefan Takacs, Numerical Analysis of Isogeometric Methods, complete JKU lecture notes, 2018/19 (standard reference, not scraped)
Annalisa Buffa, Gregor Gantner, Carlotta Giannelli, Dirk Praetorius, Rafael Vázquez, Mathematical Foundations of Adaptive Isogeometric Analysis, Archives of Computational Methods in Engineering 29 (2022) (standard reference, not scraped)