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Smooth compactly supported data have a degree-one NURBS energy rate

Statement

Assume Countable Choice. Let $F : \hat{Ω} \to D$ be one admissible NURBS patch, $\hat{Ω} = (0, 1)^{d}$ , $d \in {2, 3}$ , with determinant upper bound $j_{+}$ , derivative and inverse-derivative bound $L$ , and continuous denominator $W$ that is $C^{1}$ on each knot cell, satisfies $W \geq w_{-} > 0$ , and has $∥ \nabla W ∥_{L^{\infty}} \leq G_{W}$ . Let $g \in C^{1} (\overline{\hat{Ω}})$ have Lipschitz gradient with constant $M_{g}$ and support contained in a nonempty compact set $K ⋐ \hat{Ω}$ , and set $\hat{u} = g / W$ and $u = \hat{u} \circ F^{- 1}$ . Let the uniform tensor grid of side $h = 1 / N$ refine the geometry knot grid, let $S_{h}$ be its tensor-product degree-one $C^{0}$ numerator space, and let $V_{h} = {(s / W) \circ F^{- 1} : s \in S_{h}} \cap H_{0}^{1} (D)$ be the corresponding isogeometric trial space. Let $I_{h} g$ be the tensor Q1 nodal interpolant. If $h < dist (K, \partial \hat{Ω}) / \sqrt{d}$ , then $v_{h} = (I_{h} g / W) \circ F^{- 1}$ belongs to $V_{h}$ . If $u$ is the weak solution of an exact symmetric scalar elliptic problem with coefficient operator bound $β$ and coercivity bound $α$ as in Exact-geometry isogeometric Poisson Galerkin estimate, and $u_{h}$ its exact-form isogeometric Galerkin solution in $V_{h}$ , then $∥ u - u_{h} ∥_{H_{0}^{1} (D), \nabla} \leq \frac{β}{α} C_{F} C_{W} C_{d} h M_{g},$ where $C_{F} = \sqrt{j_{+}} \sqrt{1 + L^{2}}$ , $C_{W} = 2 / w_{-} + G_{W} / w_{-}^{2}$ , and $C_{d} = d + \sqrt{d} + 2^{d - 1} d \sqrt{d}$ . These constants retain their map, weight, coefficient, and dimension dependence. This proves only the degree-one compact-support branch; it is not a general high-degree, knot-family, or locking-free rate.

Facts & Assumptions

Given: The admissible patch and denominator bounds; the compactly supported $C^{1, 1}$ parameter numerator $g$ ; the uniform degree-one tensor grid; and the exact elliptic weak problem described in the Statement.

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ) is used by the imported Hilbert-space Céa result in [F6]; the local interpolation argument does not require the Axiom of Choice.

[F1]

Tensor Q1 nodal interpolation has a first-order H1 error bound: for the uniform tensor grid and $C^{1, 1}$ numerator, $I_{h} g$ is in the degree-one tensor spline space and $∥ g - I_{h} g ∥_{H^{1} (\hat{Ω})} \leq C_{d} h M_{g}$ .

[F2]

Compactly supported H1 functions lie in H1_0: compactly supported $H^{1}$ functions in the reference domain belong to $H_{0}^{1}$ .

[F3]

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

[F4]

NURBS quotient spaces transfer numerator H1 approximation: under Countable Choice, a continuous positive denominator and cellwise C1-up-to-boundary target and comparison with matching traces bound the physical H1 error by $C_{F} C_{W}$ times the numerator H1 error.

[F5]

Bi-Lipschitz NURBS maps preserve H1 and energy norms: the map preserves $H_{0}^{1}$ and gives the stated physical/reference norm factor $C_{F}$ .

[F6]

Exact-geometry isogeometric Poisson Galerkin estimate: for the exact symmetric form with coefficient upper and coercivity constants $β$ and $α$ , the exact-geometry Galerkin error in gradient norm is at most $β / α$ times best approximation.

Proof

1.1F1F2F3F5givenalgebra

If $g = 0$ , then $s = 0$ , $u = v_{h} = 0$ and the result is immediate. Otherwise use the nonempty support set $K$ from the Statement. Every tensor cell meeting $\partial \hat{Ω}$ has all its vertices outside $K$ : each vertex is within $\sqrt{d} h$ of a boundary point. Therefore $g$ is zero at those vertices, and the cellwise Q1 interpolant is identically zero on each such boundary cell. The function $s : = I_{h} g$ is continuous, has support in the compact $\sqrt{d} h$ -neighborhood of $K$ inside $\hat{Ω}$ , and is a degree-one tensor spline by [F1]. Both $s / W$ and $g / W$ are piecewise $C^{1}$ , have matching traces because $s, g, W$ are continuous and $W \geq w_{-} > 0$ , and have square-integrable cell gradients. By [F3] they belong to $H^{1}$ ; their supports are compact in the reference box, so [F2] puts them in $H_{0}^{1} (\hat{Ω})$ . Pullback by [F5] gives $v_{h}, u \in H_{0}^{1} (D)$ .

1.2A1F1F4givenalgebra

The grid compatibility in the Statement makes $s = I_{h} g$ an element of the numerator space defining $V_{h}$ . Apply [F4] with target numerator $g = W \hat{u}$ and approximating numerator $s$ . The numerator $g$ is C1 on the closed box, $s$ is a continuous cellwise polynomial, and the continuous denominator $W$ is a positive cellwise polynomial. Hence $g / W$ and $s / W$ are C1 up to each common-refinement cell boundary and have matching traces; Countable Choice is [A1], so all hypotheses hold and $∥ u - v_{h} ∥_{H^{1} (D)} \leq C_{F} C_{W} ∥ g - I_{h} g ∥_{H^{1} (\hat{Ω})} \leq C_{F} C_{W} C_{d} h M_{g},$ where the final inequality is [F1].

2.1A1F6step 1.2givenalgebra∎

Apply the exact-form Galerkin estimate [F6] and use $v_{h}$ as a candidate in its best-approximation infimum. The gradient error is at most $(β / α) ∥ u - v_{h} ∥_{H^{1} (D)}$ ; insert step 1.2 to obtain the displayed bound. No quadrature, perturbed geometry, or high-degree approximation rate is used.

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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)