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The convex-obstacle nodal set is conforming

Statement

Under Conforming nodal P1 Signorini contact discretization, $K_{h}$ is a nonempty closed convex subset of the finite-dimensional space $V_{h}$ and $K_{h} \subseteq K$ . If $v \in K \cap H^{2} (Ω; R^{2}) \cap C (\overline{Ω}; R^{2})$ , its global P1 nodal interpolant $I_{h} v$ is well defined, belongs to $V_{h}$ , and lies in $K_{h}$ .

Facts & Assumptions

Given: A boundary-conforming matching triangular mesh, the continuous P1 space $V_{h}$ , a continuous convex obstacle $ψ \leq 0$ on the straight segment $Γ_{C}$ , and the nodal set $K_{h}$ of Conforming nodal P1 Signorini contact discretization.

[F1]

The contact set is $K = {v \in V : T_{C} v \cdot n \geq ψ a.e. on Γ_{C}}$ and contains zero when $ψ \leq 0$ . (The vector Signorini contact set and variational inequality, The Signorini contact set is nonempty, closed, and convex)

[F2]

A matching mesh has nondegenerate triangles with shared facets equal to whole facets, and conforming P1 traces are affine on each facet. (Matching affine simplicial meshes and shape regularity, Broken and conforming polynomial spaces on a matching mesh)

[F3]

$I_{h} ψ$ is the affine interpolant of the endpoint values on each contact edge, and $K_{h}$ imposes $v_{h} (a) \cdot n \geq I_{h} ψ (a)$ at every contact vertex. (Conforming nodal P1 Signorini contact discretization)

[F4]

If an $H^{1}$ function has a representative extending continuously to each closed collar, $T_{D}$ and $T_{C}$ agree with its boundary restriction to the corresponding segments. In particular, this holds for $H^{1} \cap C (\overline{Ω})$ functions. (Partial-trace displacement space on a polygon, The $L^{p}$ trace operator on a bounded $C^{1}$ domain)

[F5]

Convexity on an interval means $ψ ((1 - t) a + t b) \leq (1 - t) ψ (a) + t ψ (b)$ for every $a, b$ in the interval and $t \in [0, 1]$ . (Convex, strictly convex, concave, strictly concave, and midpoint-convex real functions on an interval)

Proof

technique · direct

Given: The hypotheses above, with $V_{h}$ the matching continuous piecewise-affine vector space.

1.1F2F3givenalgebra

The set $V_{h}$ is finite dimensional. By [F3], $K_{h}$ is the intersection of finitely many closed affine halfspaces in $V_{h}$ , so it is closed and convex. The zero field is in $V_{h}$ , and $0 \geq I_{h} ψ (a) = ψ (a)$ at every contact vertex since $ψ \leq 0$ ; hence $0 \in K_{h}$ and $K_{h}$ is nonempty.

1.2F1F2F3F4F5givenalgebra

Fix a contact edge $e = [a, b]$ . For $v_{h} \in K_{h}$ , the function $q_{h} : = v_{h} \cdot n - I_{h} ψ$ is affine on $e$ and is nonnegative at $a, b$ by [F3]. Thus $q_{h} ((1 - t) a + t b) = (1 - t) q_{h} (a) + t q_{h} (b) \geq 0$ throughout $e$ . Convexity [F5] gives $I_{h} ψ ((1 - t) a + t b) \geq ψ ((1 - t) a + t b)$ , so $v_{h} \cdot n \geq ψ$ on every contact edge. Since $v_{h} \in C (\overline{Ω})$ , [F4] identifies its contact trace with this restriction. Therefore $v_{h} \in K$ and $K_{h} \subseteq K$ .

2.1F1F2F3F4givenconstructalgebra∎

Let $v \in K \cap H^{2} \cap C (\overline{Ω})$ . Its continuous gap $T_{C} v \cdot n - ψ$ is nonnegative almost everywhere by [F1], hence everywhere: if it were negative at one point, continuity would make it negative on a subsegment of positive length. Define $I_{h} v$ cell by cell as the affine polynomial matching $v$ at the three vertices. On a shared edge the two affine traces agree at both endpoints, so they agree along the whole edge. For every test function compactly supported in $Ω$ , integration by parts on each triangle cancels the internal-edge boundary terms because these traces agree; the remaining piecewise constant gradients are in $L^{2} (Ω)$ . Hence $I_{h} v \in H^{1} (Ω; R^{2})$ and is globally continuous. Its boundary values on $Γ_{D}$ are zero: $v \in V$ means $T_{D} v = 0$ almost everywhere, and [F4] identifies this trace with the continuous boundary restriction, which therefore vanishes everywhere on the segment. Since the mesh resolves $Γ_{D}$ , the affine boundary traces of $I_{h} v$ vanish there, so $I_{h} v \in V_{h}$ . At every contact node, $I_{h} v (a) = v (a)$ and [F4] identifies the contact trace with the continuous restriction; the continuous gap is nonnegative, so the nodal constraints hold and $I_{h} v \in K_{h}$ .

Depends on

Used by

Cited to discharge well-definedness by Conforming nodal P1 Signorini contact discretization.

Dependency tree · two levels

28 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

Richard S. Falk, Error Estimates for the Approximation of a Class of Variational Inequalities, complete 10-page paper (standard reference, not scraped)
John Andersson, The Obstacle Problem, complete 52-page KTH lecture notes (standard reference, not scraped)