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Tensorized spline quasi-interpolation gives a NURBS L2 rate

Statement

Assume the Axiom of Choice. On each coordinate interval $[0, 1]$ , let a clamped degree- $p_{r}$ analysis knot mesh have largest cell size $h_{r}$ , and let $Q_{r}$ be a family as in Bounded local spline quasi-interpolants with reproduction order $0 \leq ℓ_{r} \leq p_{r}$ , local constant $M_{r}$ , and stability constant $C_{r}$ . Require each analysis knot mesh to refine the corresponding geometry knot vector, and let $S_{h} = ⨂_{r = 1}^{d} S_{p_{r}, ξ^{(r)}}$ be the corresponding numerator space defining the patch trial space in Conforming isogeometric Galerkin spaces and h/p/k refinement. Suppose $g : \hat{Ω} \to R$ belongs to $L^{2} (\hat{Ω})$ , has weak pure derivatives $\partial_{r}^{ℓ_{r} + 1} g \in L^{2} (\hat{Ω})$ , and, for each $r$ , almost every section in coordinate $r$ belongs to $W^{ℓ_{r} + 1, 2} (0, 1)$ and satisfies the hypotheses of the one-dimensional estimate in Sobolev approximation by local spline quasi-interpolants. Require each $Q_{r}$ to have its bounded extension to $L^{2} (\hat{Ω})$ , acting in coordinate $r$ , with norm at most $S_{r} = C_{r} \sqrt{M_{r}}$ ; require the coordinate operators to commute pairwise and to commute with weak derivatives in every other coordinate on this class. These are explicit hypotheses on the supplied family, not conclusions of the univariate rate theorem. Under the Axiom of Choice, completed-product Fubini is available, and the sectionwise operators and derivatives are assumed measurable. Define $Q : = Q_{d} \dots Q_{1}$ and set $A_{r} : = (1 + C_{r}) M_{r}^{ℓ_{r} + 1 + 1 / 2} / ℓ_{r}!$ . Then $Q g \in S_{h}$ and $∥ g - Q g ∥_{L^{2} (\hat{Ω})} \leq \sum_{r = 1}^{d} A_{r} ⁣ (\prod_{j \neq r} S_{j}) h_{r}^{ℓ_{r} + 1} ∥ \partial_{r}^{ℓ_{r} + 1} g ∥_{L^{2} (\hat{Ω})} .$ If $g = W (\hat{u})$ and $v_{h} : = (Q g / W) \circ F^{- 1}$ , then $∥ u - v_{h} ∥_{L^{2} (Ω)} \leq \frac{\sqrt{j_{+}}}{w_{-}} ∥ g - Q g ∥_{L^{2} (\hat{Ω})}, u : = \hat{u} \circ F^{- 1} .$ This is an $L^{2}$ approximation estimate. It does not imply an $H^{1}$ energy rate.

Facts & Assumptions

Given: The Axiom of Choice; coordinatewise clamped meshes and quasi-interpolant families with the extension, stability, commutation, section-regularity, and measurability hypotheses of the Statement; the finite tensor-product spline space; numerator $g$ and NURBS denominator $W$ ; and map $F$ with Jacobian upper bound $j_{+}$ .

[A1]

The Axiom of Choice is assumed for the one-dimensional Sobolev quasi-interpolation estimate and completed-product Fubini used below; it is not inferred from the weaker Countable Choice branch used by the pullback theorem.

[F1]

Bounded local spline quasi-interpolants: each $Q_{r}$ is linear, local, reproduces all polynomials through degree $ℓ_{r}$ , and obeys the common local $L^{2}$ stability bound with overlap at most $M_{r}$ .

[A2]

In addition to [F1], the Statement assumes bounded extensions to $L^{2} (\hat{Ω})$ with norm $S_{r}$ , pairwise commutation, commutation with weak derivatives in other coordinates on the anisotropic class, and the stated section-regularity and measurability.

[F2]

Sobolev approximation by local spline quasi-interpolants under the Axiom of Choice: for a fixed clamped mesh, $∥ f - Q_{r} f ∥_{L^{2}} \leq A_{r} h_{r}^{ℓ_{r} + 1} ∥ f^{(ℓ_{r} + 1)} ∥_{L^{2}}$ with the displayed $A_{r}$ .

[F3]

Tonelli and Fubini for the completed product, with only almost-everywhere section measurability: nonnegative section integrals over a completed product agree with the iterated integrals. The Euclidean Lebesgue measure is the completion of the product of the factor Lebesgue measures identifies the Euclidean box measure with this completed product.

[F4]

Conforming isogeometric Galerkin spaces and h/p/k refinement defines the analysis space as $S_{h} / W$ for analysis knots refining the geometry grid; Positive weights give a bounded partition-of-unity NURBS basis gives $W \geq w_{-} > 0$ .

[F5]

A C^1 diffeomorphism satisfies the change-of-variables formula for nonnegative Lebesgue measurable functions gives substitution for every nonnegative measurable integrand on each open cell diffeomorphism under Countable Choice. Additivity of the nonnegative Lebesgue integral sums these integrals over the finite partition; the admissible reference and physical cell faces are null. The choice implication AC supplies the countable and dependent choices used in Banach integration supplies Countable Choice under [A1].

Proof

1.1A1A2F1F2F3givenalgebra

For a continuous one-dimensional section $f$ , square the local stability inequalities in [F1] and sum over the disjoint knot cells. Each point belongs to at most $M_{r}$ neighborhoods, so $∥ Q_{r} f ∥_{L^{2} (0, 1)}^{2} \leq C_{r}^{2} M_{r} ∥ f ∥_{L^{2} (0, 1)}^{2}$ and therefore $∥ Q_{r} f ∥_{2} \leq S_{r} ∥ f ∥_{2}$ . Apply [F2] to almost every section and integrate the squared estimates over the remaining coordinates by [F3]. This gives $∥ (I - Q_{r}) g ∥_{L^{2} (\hat{Ω})} \leq A_{r} h_{r}^{ℓ_{r} + 1} ∥ \partial_{r}^{ℓ_{r} + 1} g ∥_{L^{2} (\hat{Ω})}$ .

2.1A2F1F3step 1.1givenalgebra

By [A2], each coordinate operator is bounded on $L^{2} (\hat{Ω})$ and the coordinate operators commute pairwise. Each $Q_{j}$ reproduces the constant function one by [F1], whose norm on the unit box is one. Its operator bound therefore gives $1 = ∥ Q_{j} 1 ∥_{2} \leq S_{j}$ , so $S_{j} \geq 1$ . Use the finite telescoping identity $I - Q_{d} \dots Q_{1} = \sum_{r = 1}^{d} Q_{d} \dots Q_{r + 1} (I - Q_{r})$ , with the product empty for $r = d$ . Step 1.1 and boundedness of the left factors bound its $r$ th summand in $L^{2}$ by $A_{r} h_{r}^{ℓ_{r} + 1} \prod_{j > r} S_{j} ∥ \partial_{r}^{ℓ_{r} + 1} g ∥_{2}$ . Since each omitted $S_{j} \geq 1$ , this is at most $A_{r} h_{r}^{ℓ_{r} + 1} \prod_{j \neq r} S_{j} ∥ \partial_{r}^{ℓ_{r} + 1} g ∥_{2}$ . Summing proves the displayed tensor-product estimate; the range is the tensor-product spline numerator space by construction.

3.1A1F4F5step 2.1algebra∎

By [F4], $Q g / W$ is a NURBS analysis function. Put $z : = (g - Q g) / W$ , so $u - v_{h} = z \circ F^{- 1}$ and $∥ z ∥_{2} \leq w_{-}^{- 1} ∥ g - Q g ∥_{2}$ . For each reference open cell $K$ , [F5] applied to $∣ z \circ F^{- 1} ∣^{2}$ on $F (K)$ gives $\int_{F (K)} ∣ z \circ F^{- 1} ∣^{2} d x = \int_{K} ∣ z (ξ) ∣^{2} J_{F} (ξ) d ξ$ . These compositions are measurable on each cell by the same change-of-variables interface; arbitrary values on the finitely many null faces do not affect their classes. Summing over the cells and using $J_{F} \leq j_{+}$ yields $∥ u - v_{h} ∥_{2}^{2} \leq j_{+} ∥ z ∥_{2}^{2} \leq j_{+} w_{-}^{- 2} ∥ g - Q g ∥_{2}^{2}$ . Taking square roots proves the physical bound for arbitrary $L^{2}$ numerator errors. No gradient estimate is used, so no energy-norm conclusion follows.

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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)
Tom Lyche, Carla Manni, Hendrik Speleers, B-Splines and Spline Approximation, complete lecture notes (standard reference, not scraped)