LemmaStatement: Literature-sourcedProof: AI-generatedPipeline-generated

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AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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A bounded form defect preserves discrete coercivity

Statement

Let $V$ be a Hilbert space and let $V_{h} \subseteq V$ be a subspace. Suppose the exact form $a$ is coercive with constant $α > 0$ in the norm $∥ \cdot ∥_{V}$ . Let $a_{h}$ be a discrete form on $V_{h}$ and suppose

$∣ a_{h} (v_{h}, v_{h}) - a (v_{h}, v_{h}) ∣ \leq η_{h} ∥ v_{h} ∥_{V}^{2} (v_{h} \in V_{h})$

for some $η_{h} \geq 0$ . Then $a_{h}$ is coercive with constant at least $α - η_{h}$ whenever $η_{h} < α$ . In particular, if $η_{h} \leq α / 2$ , then its coercivity constant is at least $α / 2$ .

Facts & Assumptions

Given: A Hilbert space $V$ , a subspace $V_{h}$ , a coercive form $a$ with constant $α > 0$ , a discrete form $a_{h}$ , and the diagonal defect bound with constant $η_{h}$ .

[F1]

Bounded, coercive and symmetric sesquilinear forms: coercivity means $Re a (v, v) \geq α ∥ v ∥_{V}^{2}$ ; symmetry is not required.

[A1]

For every $v_{h} \in V_{h}$ , $∣ a_{h} (v_{h}, v_{h}) - a (v_{h}, v_{h}) ∣ \leq η_{h} ∥ v_{h} ∥_{V}^{2}$ .

Proof

1.1givenF1A1algebra

Fix $v_{h} \in V_{h}$ . By coercivity [F1] and the defect hypothesis [A1], $Re a_{h} (v_{h}, v_{h}) = Re a (v_{h}, v_{h}) + Re (a_{h} (v_{h}, v_{h}) - a (v_{h}, v_{h})) \geq (α - η_{h}) ∥ v_{h} ∥_{V}^{2}$ .

2.1step 1.1algebra∎

If $η_{h} < α$ , then $α - η_{h} > 0$ , so the inequality in step 1.1 is precisely coercivity with that constant. If $η_{h} \leq α / 2$ , then $α - η_{h} \geq α / 2$ .

Depends on

Used by

Cellwise geometry tensor gaps bound form error and stability loss Lemma

Dependency tree · two levels

8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

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, Chapter 25, Non-conforming Finite Element Methods (standard reference, not scraped)