TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck pass

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Mixed Galerkin error bound retaining beta_h

Statement

Assume Countable Choice. Let $V, Q$ be real Hilbert spaces and let $F \in V'$ , $G \in Q'$ be bounded data. Suppose bounded bilinear forms $a, b$ and $(u, p) \in V \times Q$ satisfy $a (u, v) + b (v, p) = F (v), b (u, q) = G (q)$ for all $v \in V, q \in Q$ . Let $V_{h} \subset V$ , $Q_{h} \subset Q$ be finite-dimensional conforming subspaces, with $V_{h} \neq {0}$ , $Q_{h} \neq {0}$ , and discrete inf-sup constant $β_{h} : = \inf_{q_{h} \neq 0} \sup_{v_{h} \neq 0} b (v_{h}, q_{h}) / (∥ v_{h} ∥_{V} ∥ q_{h} ∥_{Q}) > 0$ . Assume the discrete kernel coercivity $a (v_{h}, v_{h}) \geq α_{*} ∥ v_{h} ∥_{V}^{2}$ for every $v_{h} \in Z_{h} : = {w_{h} : b (w_{h}, q_{h}) = 0 for all q_{h} \in Q_{h}}$ , where $α_{*} > 0$ . Put $M_{a} = ∥ a ∥$ , $M_{b} = ∥ b ∥$ . The Galerkin system with the same $F, G$ has a unique solution $(u_{h}, p_{h})$ . For any $w_{h} \in V_{h}$ , $r_{h} \in Q_{h}$ , define $δ_{u} = ∥ u - w_{h} ∥_{V}$ , $δ_{p} = ∥ p - r_{h} ∥_{Q}$ , $R_{h} = β_{h}^{- 1} M_{b} δ_{u}$ , and $U_{h} = δ_{u} + α_{*}^{- 1} {M_{a} (δ_{u} + R_{h}) + M_{b} δ_{p}} + R_{h},$ $P_{h} = δ_{p} + β_{h}^{- 1} {M_{a} (U_{h} + δ_{u}) + M_{b} δ_{p}} .$ Then $∥ u - u_{h} ∥_{V} \leq U_{h}$ and $∥ p - p_{h} ∥_{Q} \leq P_{h}$ . In particular the pressure bound retains $β_{h}^{- 1}$ explicitly; taking infima over $w_{h}, r_{h}$ gives the corresponding best-approximation bound.

Facts & Assumptions

Given: A continuous solution, nonzero conforming finite-dimensional subspaces $V_{h}, Q_{h}$ , positive discrete inf-sup $β_{h}$ , boundedness constants $M_{a}, M_{b}$ , and coercivity on $Z_{h}$ with constant $α_{*}$ . The discrete equations use the same $F, G$ .

[A1]

Countable Choice is used through the Babuška–Brezzi/Lax–Milgram solvability theorem for the discrete pair. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[F1]

On any real Hilbert pair, coercivity on the constraint kernel and a positive inf-sup constant give unique mixed solvability. (Babuska-Brezzi solvability by kernel coercivity and inf-sup)

[F2]

The discrete inf-sup bound gives a right inverse of the discrete constraint operator with norm at most $β_{h}^{- 1}$ . (Hilbert inf-sup gives a bounded constraint right inverse)

Proof

1.1A1F1given

Apply [F1] to $V_{h}, Q_{h}$ with the assumed $Z_{h}$ -coercivity and discrete inf-sup constant $β_{h}$ . Since the restricted forms are bounded, the Galerkin system with the same $F, G$ has a unique solution $(u_{h}, p_{h})$ .

2.1A1givenstep 1.1

Fix $w_{h}, r_{h}$ and set $θ_{u} = w_{h} - u_{h}$ , $θ_{p} = r_{h} - p_{h}$ , $η_{u} = u - w_{h}$ , $η_{p} = p - r_{h}$ . Subtracting the discrete equations from the continuous equations gives $a (θ_{u}, v_{h}) + b (v_{h}, θ_{p}) = - a (η_{u}, v_{h}) - b (v_{h}, η_{p})$ and $b (θ_{u}, q_{h}) = - b (η_{u}, q_{h})$ .

3.1A1F2step 2.1

The second identity implies $∥ B_{h} θ_{u} ∥_{Q} \leq M_{b} δ_{u}$ . By [F2] choose $r_{u} \in V_{h}$ with $B_{h} r_{u} = B_{h} θ_{u}$ and $∥ r_{u} ∥_{V} \leq R_{h}$ . Then $z_{h} = θ_{u} - r_{u} \in Z_{h}$ .

4.1givenstep 2.1step 3.1

Test the first identity of step 2.1 with $z_{h}$ . Since $b (z_{h}, θ_{p}) = 0$ , $a (z_{h}, z_{h}) = - a (η_{u}, z_{h}) - b (z_{h}, η_{p}) - a (r_{u}, z_{h})$ . Discrete-kernel coercivity and boundedness imply, when $z_{h} \neq 0$ , $∥ z_{h} ∥ \leq α_{*}^{- 1} {M_{a} (δ_{u} + R_{h}) + M_{b} δ_{p}}$ ; if $z_{h} = 0$ this bound is immediate. Therefore $∥ u - u_{h} ∥ \leq ∥ η_{u} ∥ + ∥ z_{h} ∥ + ∥ r_{u} ∥ \leq U_{h}$ .

5.1A1F2algebrastep 2.1step 4.1∎

Since $θ_{u} + η_{u} = u - u_{h}$ , the first identity of step 2.1 gives $b (v_{h}, θ_{p}) = - a (u - u_{h}, v_{h}) - b (v_{h}, η_{p})$ . Therefore $∣ b (v_{h}, θ_{p}) ∣ \leq [M_{a} U_{h} + M_{b} δ_{p}] ∥ v_{h} ∥$ by step 4.1. Taking the supremum and applying the discrete inf-sup yields $∥ θ_{p} ∥ \leq β_{h}^{- 1} (M_{a} U_{h} + M_{b} δ_{p}) \leq β_{h}^{- 1} {M_{a} (U_{h} + δ_{u}) + M_{b} δ_{p}}$ . Hence $∥ p - p_{h} ∥ \leq δ_{p} + ∥ θ_{p} ∥ \leq P_{h}$ .

Depends on

Used by

Dependency tree · two levels

11 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, Chapters 49–54 (complete course notes) (standard reference, not scraped)
Daniele Boffi, Franco Brezzi, Michel Fortin, Mixed Finite Element Methods and Applications (complete textbook) (standard reference, not scraped)