Localization of strict half-relaxed contacts to periodic mesh nodes

Choose a closed ball $K$ about $z_0$ contained in the open cylinder where the contact is strict. For any smaller radius $r>0$, the annulus $K\setminus B_r(z_0)$ is compact by [F3]; upper semicontinuity of $\bar{V}-\varphi$ makes its maximum strictly less than $\bar{V}(z_0)-\varphi (z_0)$. Denote the positive gap by $g_r$. The lower contact case gives the corresponding positive gap above the strict minimum.

By [F1] and [A1], select pairs $(k_m,y_m)$ with $k_m\to \mathrm{\infty }$, $y_m\to z_0$, and $V_{k_m}(y_m)-\varphi (y_m)\to \bar{V}(z_0)-\varphi (z_0)$. The interpolated value $V_{k_m}(y_m)$ is a convex combination of the nodal values in its cell by [F2], so at least one cell vertex has value no smaller. Choose the first such vertex in the lexicographic order on its binary cell label $ϵ\in \{0,1{\}}^d$, where at the first differing coordinate $0$ precedes $1$. Its spatial distance from $y_m$ is at most $\sqrt{d}{h}_{k_m}$; the piecewise-constant time extension has a grid time within ${\tau }_{k_m}$. Thus the selected nodal points converge to $z_0$. Continuity of $\varphi$ makes the nodal $V-\varphi$ values at least the recovery values minus a quantity tending to zero, so their lower limit is at least the upper contact value.

For all sufficiently large $m$, maximize the finite nonempty set of nodal values $U_i^n-\varphi (x_i,t_n)$ inside $K$ and take its lexicographically first maximizing tuple $(i,n)$, comparing integer coordinates in displayed order and putting the smaller first differing coordinate first. A maximizer exists by [F3]; its value is at least the recovery value from step 1.2. Fix $r>0$ and let $A_r=K\setminus B_r(z_0)$. The strict gap $g_r$ from step 1.1 holds on $A_r$. For every $z\in A_r$, the set-based definition [F1] supplies a positive radius on which $V_{ϵ}(y)\le \bar{V}(z)+g_r\mathrm{/}8$ whenever both $ϵ$ and $\mathrm{\mid }y-z\mathrm{\mid }$ are smaller than that radius. Continuity of $\varphi$ and density of the positive rationals allow the following predicate on pairs $(z,q)$: $P_r(z,q)$ means $V_{ϵ}(y)\le \bar{V}(z)+g_r\mathrm{/}8$ for every $0<ϵ<q$ and $\mathrm{\mid }y-z\mathrm{\mid }<q$, and $\mathrm{\mid }\varphi (y)-\varphi (z)\mathrm{\mid }<g_r\mathrm{/}8$ for every $\mathrm{\mid }y-z\mathrm{\mid }<q$. The family ${\mathcal{U}}_r=\{B(z,q\mathrm{/}2):z\in A_r,q\in {\mathbb{Q}}_{>0},P_r(z,q)\}$ is an open cover of $A_r$: for each center some real radius works by the relaxed-limit definition and continuity, and a smaller rational radius also works. This is the set of all qualifying rational pairs, not a choice of one radius for each $z$. By [F3] it has a finite subcover; let $q_{\ast }>0$ be the least radius in that finite subcover. Whenever $ϵ<q_{\ast }$ and $y\in A_r$, a covering center gives $V_{ϵ}(y)-\varphi (y)\le \bar{V}(z)-\varphi (z)+g_r\mathrm{/}4\le \bar{V}(z_0)-\varphi (z_0)-3g_r\mathrm{/}4.$ The recovery value in step 1.2 is eventually larger than this bound, so no maximizer on the grid in $K$ lies in $A_r$. Since this holds for every $r>0$, the maximizing nodes converge to $z_0$. Their values have lower limit at least the contact value by step 1.2; the upper-limit definition [F1], applied at these nodes, gives upper limit at most the same value. Thus their values converge to the upper half-relaxed contact value.

For the lower half-relaxed limit, [F1] and [A1] select near-recovery pairs with values approaching $\underset{‾}{V}(z_0)$. Convex interpolation supplies a cell vertex with value no larger than the interpolated value; take the first such vertex in the binary-cell lexicographic order. Minimize over the finite grid points in $K$ and take the lexicographically first minimizer. Fix $r>0$ and use the strict lower-contact gap $g_r$ from step 1.1 on $A_r$. For each $z\in A_r$, the set-defined lower limit gives a positive radius on which $V_{ϵ}(y)\ge \underset{‾}{V}(z)-g_r\mathrm{/}8$ whenever both $ϵ$ and $\mathrm{\mid }y-z\mathrm{\mid }$ are smaller than that radius. Define $P_r^{-}(z,q)$ to mean this bound holds for every $0<ϵ<q$ and $\mathrm{\mid }y-z\mathrm{\mid }<q$, and $\mathrm{\mid }\varphi (y)-\varphi (z)\mathrm{\mid }<g_r\mathrm{/}8$ for every $\mathrm{\mid }y-z\mathrm{\mid }<q$. The family of all $B(z,q\mathrm{/}2)$ with $z\in A_r$, rational $q>0$, and $P_r^{-}(z,q)$ covers $A_r$; as in step 2.1, it uses all qualifying pairs rather than a pointwise choice. A finite subcover from [F3] has least radius $q_{\ast }>0$, so every sufficiently fine approximation at every $y\in A_r$ satisfies $V_{ϵ}(y)-\varphi (y)\ge \underset{‾}{V}(z)-\varphi (z)-g_r\mathrm{/}4\ge \underset{‾}{V}(z_0)-\varphi (z_0)+3g_r\mathrm{/}4.$ The lower recovery values from the selected vertex tend to $\underset{‾}{V}(z_0)-\varphi (z_0)$; hence no minimizing node lies in $A_r$ once the mesh is sufficiently fine. As this holds for each $r>0$, the minimizing nodes approach $z_0$. Their upper limit is at most the recovery contact value, while the set-defined lower limit [F1] gives lower limit at least that value, so they converge to it. This proves the dual statement.