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Exact-geometry isogeometric Poisson Galerkin estimate

Statement

Assume Countable Choice. Let $D \subset R^{d}$ , $d \in {2, 3}$ , be bounded, connected, and Lipschitz, and let $V = H_{0}^{1} (D; R)$ with norm $∥ v ∥_{V} = ∥ \nabla v ∥_{L^{2} (D)}$ . Let $A : D \to R^{d \times d}$ be measurable and symmetric, with constants $0 < α \leq β < \infty$ such that $ζ \cdot A (x) ζ \geq α ∣ ζ ∣^{2} and ∥ A (x) ∥_{o p} \leq β for a.e. x \in D, ζ \in R^{d} .$ For $f \in V^{'}$ , define $a (u, v) = \int_{D} A \nabla u \cdot \nabla v$ and $F (v) = f (v)$ . Let $V_{h} \subset V$ be any finite conforming isogeometric subspace on the exact admissible patch geometry. Then the weak problem and its Galerkin restriction have unique solutions $u$ and $u_{h}$ , and $∥ u - u_{h} ∥_{V} \leq \frac{β}{α} \inf_{v_{h} \in V_{h}} ∥ u - v_{h} ∥_{V} .$ This is an exact-form estimate; it makes no assertion about approximate geometry, quadrature, polynomial-degree-independent constants, or a rate without a separate numerator approximation theorem.

Facts & Assumptions

Given: Countable Choice; the bounded Lipschitz domain and scalar Sobolev space above; measurable symmetric coefficient $A$ with the stated bounds; $f \in V^{'}$ ; and finite conforming isogeometric $V_{h} \subset V$ .

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ) is used through the Hilbert-space and Céa suppliers in [F3]--[F4]; no stronger choice principle is required by this application.

[F1]

Conforming isogeometric Galerkin spaces and h/p/k refinement: the exact-geometry global trial space is finite dimensional, and its homogeneous-Dirichlet version is by definition a subspace of $H_{0}^{1} (D)$ .

[F2]

Matching NURBS patch traces give global H1 conformity proves global $H^{1}$ membership from matching full-face traces for the piecewise smooth fields used to construct multipatch spaces.

[F3]

The Sobolev space $H^{1}$ is a Hilbert space and Zero-boundary Sobolev space as a norm closure: $H_{0}^{1} (D)$ is closed in the $H^{1}$ Hilbert space under Countable Choice. The bounded-domain Poincaré inequality A bounded-domain Poincare inequality for H1-zero fields makes $∥ \nabla \cdot ∥_{2}$ an equivalent Hilbert norm.

[F4]

Céa quasi-optimality for coercive conforming Galerkin problems under Countable Choice: for a Hilbert space, a bounded form constant $M$ , coercivity constant $α_{0} > 0$ , bounded data, and a finite-dimensional subspace, the continuous and discrete solutions are unique and the error is at most $(M / α_{0})$ times best approximation.

Proof

1.1givenalgebra

The operator bound on $A$ and Cauchy--Schwarz give $∣ a (v, w) ∣ \leq β ∥ v ∥_{V} ∥ w ∥_{V}$ . The lower coefficient bound gives $a (v, v) \geq α ∥ v ∥_{V}^{2}$ . Hence $a$ is bounded with $M = β$ and coercive with $α_{0} = α$ .

2.1A1F1F2F3F4givenstep 1.1∎

By [F3], $V$ with the gradient norm is a real Hilbert space. Its data functional $F$ is bounded by the definition of $V^{'}$ , and $V_{h}$ is finite dimensional and conforming by [F1] (with [F2] verifying the interface condition when needed). Thus all hypotheses of [F4] hold with $M = β$ and $α_{0} = α$ . Applying [F4] gives both solutions and its estimate is exactly the displayed inequality. Its hypotheses concern the exact form only, so the conclusion carries no geometry or quadrature defect term and no approximation rate beyond the best-approximation infimum.

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Sources

Jean-Luc Guermond, MATH 661 Fall 2025, Part VI, Chapter 26, Basic error analysis (standard reference, not scraped)
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)