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GLL spectral-element energy error retains the quadrature defect

Statement

Assume Countable Choice. Let $Ω$ have a fixed finite matching affine tensor-box mesh with cell degrees $p_{K} \geq 1$ , and let $V_{p}$ be its trace-matching hp space. On $V = H^{1} (Ω)$ use $a (v, w) = \int_{Ω} (\nabla v \cdot \nabla w + v w)$ and a bounded load $F$ . Define the discrete form on $V_{p}$ by

a_{p} (v_{p}, w_{p}) = \sum_{K \in T} (a_{K, p_{K}} (v_{p} ∣_{K}, w_{p} ∣_{K}) + m_{K, p_{K}} (v_{p} ∣_{K}, w_{p} ∣_{K})),

using the GLL stiffness and mass forms from Legendre-Gauss-Lobatto nodes, weights, and tensor quadrature. Let $u$ solve the exact variational problem and let $u_{p} \in V_{p}$ solve $a_{p} (u_{p}, w_{p}) = F (w_{p})$ for every $w_{p} \in V_{p}$ , with the exact load restricted to $V_{p}$ . Define the form defect $δ_{a, p}$ by Variational-crime settings and consistency defects. Then $∥ u - u_{p} ∥_{H^{1} (Ω)} \leq \inf_{v_{p} \in V_{p}} (2 ∥ u - v_{p} ∥_{H^{1} (Ω)} + δ_{a, p} (v_{p})) .$ The stability factor is uniform in all cell degrees. The defect is retained as stated; no aliasing or p-decay estimate is asserted, and GLL assembly is not identified with exact Galerkin assembly.

Facts & Assumptions

Given: Countable Choice; the finite matching affine tensor-box mesh and $p_{K} \geq 1$ ; exact $a, F, u$ ; the GLL form $a_{p}$ ; and its discrete solution $u_{p}$ .

[F1]

Matching variable-degree polynomial traces give an H1 function: the trace-matching hp space is finite-dimensional and contained in $H^{1} (Ω)$ .

[F2]

GLL quadrature changes only the top Legendre modes and is p-uniformly equivalent: on each affine box the GLL quadrature H1 form is bounded above by $3^{d}$ times and below by $1$ times the exact local H1 form, uniformly in its degree.

[F3]

Variational-crime settings and consistency defects: $δ_{a, p} (v_{p}) = \sup_{w_{p} \neq 0} ∣ a (v_{p}, w_{p}) - a_{p} (v_{p}, w_{p}) ∣ / ∥ w_{p} ∥_{H^{1}}$ , while the load defect is defined analogously.

[F4]

First Strang estimate for conforming perturbed Galerkin problems: under Countable Choice, a coercive exact and discrete form on a finite-dimensional conforming subspace satisfy the first Strang estimate with the form and load defects from [F3].

[F5]

Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs: a positive weighted finite sum obeys Cauchy–Schwarz.

[F6]

Legendre-Gauss-Lobatto nodes, weights, and tensor quadrature defines the cell stiffness form $a_{K, p_{K}}$ and mass form $m_{K, p_{K}}$ used in the displayed discrete form.

Proof

1.1F1F2F3F5F6givenalgebra

Sum [F2] over cells. For every $v_{p} \in V_{p}$ , $a (v_{p}, v_{p}) = ∥ v_{p} ∥_{H^{1}}^{2}$ and $a_{p} (v_{p}, v_{p}) \geq ∥ v_{p} ∥_{H^{1}}^{2}$ ; Cauchy–Schwarz [F5] for the positive quadrature weights gives $∣ a_{p} (v_{p}, w_{p}) ∣ \leq 3^{d} ∥ v_{p} ∥_{H^{1}} ∥ w_{p} ∥_{H^{1}}$ . Thus the discrete form is bounded and coercive with $α_{p} \geq 1$ , and the exact form has $M = 1$ . The exact load is unchanged on $V_{p}$ , so $δ_{F, p} = 0$ .

2.1F1F3F4step 1.1givenalgebra∎

Apply [F4] with $V = H^{1} (Ω)$ , the finite-dimensional conforming subspace from [F1], exact bounded coercive form $a$ , GLL form $a_{p}$ , and the restricted load. Its bound is $\inf_{v_{p} \in V_{p}} ((1 + M / α_{p}) ∥ u - v_{p} ∥ + (δ_{a, p} (v_{p}) + δ_{F, p}) / α_{p})$ . Substituting $M = 1$ , $α_{p} \geq 1$ , and $δ_{F, p} = 0$ from step 1.1 yields the stated inequality. Countable Choice is the only axiom used by [F4].

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Sources

Jean-Luc Guermond, MATH 661 Fall 2025, Part VI, Chapter 27, Error analysis with variational crimes (standard reference, not scraped)
Joachim Schöberl, Numerical Methods for Partial Differential Equations (standard reference, not scraped)