TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck 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.

The dual mortar contact system is stable and exactly complementary

Statement

On each fixed contact mesh in Dual P1 mortar multipliers for boundary contact, assume $K_{h} \subseteq K$ , the elasticity form is bounded and coercive on $V_{h}$ with constants $M, α > 0$ , and $F$ is bounded. Then the nodal contact variational inequality has a unique displacement $u_{h}$ and a unique multiplier $λ_{h} = \sum_{i} λ_{i} χ_{i} \in Λ_{h}^{+}$ satisfying $a (u_{h}, v_{h}) - F (v_{h}) = b_{h} (v_{h}, λ_{h}) (v_{h} \in V_{h}),$ $g_{i} : = u_{h} (a_{i}) \cdot n - I_{h} ψ (a_{i}) \geq 0, λ_{i} \geq 0, λ_{i} g_{i} = 0 (0 \leq i \leq N) .$ These saddle equations are equivalent to the discrete VI, and the integrated complementarity residual is exactly $R_{h} : = \int_{Γ_{C}} λ_{h} (T_{C} u_{h} \cdot n - I_{h} ψ) d s = \sum_{i} d_{i} λ_{i} g_{i} = 0$ . The trace-pair inf-sup bound is at least $1 / \sqrt{3}$ in the stated $L^{2}$ norms by The dual mortar basis is biorthogonal and uniformly stable on the trace space, and the displacement satisfies $∥ u_{h} ∥_{V} \leq ∥ F ∥_{V^{*}} / α$ . The coupling uses exact edge integrals and has no contact quadrature defect.

Facts & Assumptions

Given: The finite-dimensional P1 displacement space $V_{h}$ , nonempty closed convex nodal set $K_{h}$ , contact nodes $a_{i}$ , edge weights $d_{i} > 0$ , exact mortar pairing $b_{h}$ , and coercive elasticity form.

[A1]

Countable Choice is used through the discrete Stampacchia theorem. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[F1]

$V_{h}$ is finite dimensional, hence Hilbert with the inherited $H^{1}$ inner product; $K_{h}$ is nonempty, closed, and convex. (Conforming nodal P1 Signorini contact discretization, The convex-obstacle nodal set is conforming, Every finite-dimensional normed space is Banach)

[F2]

The discrete form is bounded by $M$ and coercive with constant $α > 0$ inherited from $V$ . (The isotropic contact elasticity form is well defined and bounded, Korn coercivity with a partial Dirichlet segment)

[F3]

The local dual basis satisfies $\int χ_{i} ϕ_{j} = d_{i} δ_{i j}$ , has trace-pair inf-sup at least $1 / \sqrt{3}$ , and $b_{h} (v_{h}, λ_{h}) = \sum_{i} d_{i} λ_{i} (v_{h} (a_{i}) \cdot n)$ . (The dual mortar basis is biorthogonal and uniformly stable on the trace space)

[F4]

For $v_{h} \in K_{h}$ , each nodal gap $v_{h} (a_{i}) \cdot n - I_{h} ψ (a_{i})$ is nonnegative, and $K_{h}$ is defined by these coordinate constraints. (Conforming nodal P1 Signorini contact discretization, The convex-obstacle nodal set is conforming)

[F5]

The coercive variational inequality on a Hilbert space with nonempty closed convex feasible set has exactly one solution. (Stampacchia's variational inequality)

[F6]

The contact-coordinate map $B_{i} (v_{h}) = v_{h} (a_{i}) \cdot n$ is onto $R^{N + 1}$ ; for each $i$ a test function $w_{i} \in V_{h}$ has $B_{j} (w_{i}) = δ_{i j}$ . (Contact nodal coordinates are independently attainable)

Proof

technique · direct

Given: $V_{h}$ , $K_{h}$ , $a$ , $F$ , the nodal trace map $B_{i} (v_{h}) : = v_{h} (a_{i}) \cdot n$ , and the dual basis pairing.

1.1A1F1F2F5F6given

By [F1]–[F2], the restricted form and load meet the hypotheses of [F5], so the discrete VI has exactly one solution $u_{h} \in K_{h}$ . By [F6], the contact-coordinate map $B : V_{h} \to R^{N + 1}$ is onto; fix test functions $w_{i}$ with $B_{j} (w_{i}) = δ_{i j}$ .

2.1F4givenalgebrastep 1.1

Define the residual $r_{h} (v_{h}) : = a (u_{h}, v_{h}) - F (v_{h})$ . If $k \in \ker B$ , then $u_{h} \pm t k \in K_{h}$ for every real $t$ , so the VI tested with both signs gives $r_{h} (k) = 0$ . If $g_{i} > 0$ , then $u_{h} \pm t w_{i} \in K_{h}$ for all sufficiently small $t > 0$ , so $r_{h} (w_{i}) = 0$ ; if $g_{i} = 0$ , the feasible variation $u_{h} + t w_{i}$ for $t > 0$ gives $r_{h} (w_{i}) \geq 0$ . Set $λ_{i} : = r_{h} (w_{i}) / d_{i}$ . Then $λ_{i} \geq 0$ and $λ_{i} g_{i} = 0$ .

3.1F3step 1.1step 2.1algebra

For arbitrary $v_{h} \in V_{h}$ , write $v_{h} = \sum_{i} B_{i} (v_{h}) w_{i} + k$ with $k : = v_{h} - \sum_{i} B_{i} (v_{h}) w_{i} \in \ker B$ . Step 2.1 gives $r_{h} (v_{h}) = \sum_{i} B_{i} (v_{h}) r_{h} (w_{i}) = \sum_{i} d_{i} λ_{i} B_{i} (v_{h}) = b_{h} (v_{h}, λ_{h})$ by [F3], with $λ_{h} : = \sum_{i} λ_{i} χ_{i} \in Λ_{h}^{+}$ . Thus the saddle stationarity equation holds. If another multiplier gives the same stationarity, surjectivity of $B$ and $d_{i} > 0$ force each coefficient to equal $λ_{i}$ , proving multiplier uniqueness.

4.1F3F4step 3.1algebra

Conversely, suppose a feasible displacement and nonnegative coefficients satisfy stationarity and $λ_{i} g_{i} = 0$ . For $w_{h} \in V_{h}$ , define its nodal gap $g_{i} (w_{h}) : = B_{i} (w_{h}) - I_{h} ψ (a_{i})$ , so $g_{i} = g_{i} (u_{h})$ . For any feasible $v_{h}$ , biorthogonality [F3] gives $a (u_{h}, v_{h} - u_{h}) - F (v_{h} - u_{h}) = \sum_{i} d_{i} λ_{i} (B_{i} (v_{h}) - B_{i} (u_{h})) = \sum_{i} d_{i} λ_{i} (g_{i} (v_{h}) - g_{i} (u_{h})) = \sum_{i} d_{i} λ_{i} g_{i} (v_{h}) \geq 0$ , since $g_{i} (u_{h}) = 0$ whenever $λ_{i} > 0$ and $g_{i} (v_{h}) \geq 0$ by [F4]. Hence the saddle system is equivalent to the VI, and $R_{h} = \sum_{i} d_{i} λ_{i} g_{i} = 0$ exactly.

5.1F1F2F3F4step 1.1algebra∎

By [F4], $ψ \leq 0$ and the nodal interpolant satisfies $I_{h} ψ (a_{i}) = ψ (a_{i})$ , so the zero displacement satisfies every defining inequality of $K_{h}$ and $0 \in K_{h}$ . Taking $v_{h} = 0$ in the discrete VI gives $a (u_{h}, u_{h}) \leq F (u_{h}) \leq ∥ F ∥_{V^{*}} ∥ u_{h} ∥_{V}$ . Coercivity [F2] yields $α ∥ u_{h} ∥_{V}^{2} \leq ∥ F ∥_{V^{*}} ∥ u_{h} ∥_{V}$ , so either $u_{h} = 0$ or division gives $∥ u_{h} ∥_{V} \leq ∥ F ∥_{V^{*}} / α$ . The trace inf-sup bound in [F3] and exact edge-integral definition of $b_{h}$ establish the stated discrete stability and consistency.

Depends on

Used by

A two-gap contact model changes its active set with the load Example

Dependency tree · two levels

54 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

Rolf H. Krause and Barbara I. Wohlmuth, Multigrid Methods for Mortar Finite Elements, complete 7-page paper (standard reference, not scraped)