Conforming nodal P1 Signorini contact discretization

Let $({\mathcal{T}}_h)$ be a family of matching, boundary-conforming, shape-regular affine triangular meshes of the polygon from Partial-trace displacement space on a polygon, with $h:={\max }_{K\in {\mathcal{T}}_h}\mathrm{diam}(K)$. Assume both ${\mathrm{\Gamma }}_D$ and ${\mathrm{\Gamma }}_C$ are unions of boundary edges of each mesh, and let ${\mathcal{N}}_h^C$ be the vertices on ${\mathrm{\Gamma }}_C$. The conforming continuous P1 displacement space is $V_h:=\{v_h\in V\cap C(\bar{\mathrm{\Omega }};{\mathbb{R}}^2):v_h{\mathrm{\mid }}_K\in {\mathbb{P}}_1(K;{\mathbb{R}}^2)\text{ for every }K\in {\mathcal{T}}_h\}\mathrm{.}$ For continuous $\psi$ on ${\mathrm{\Gamma }}_C$, let $I_h\psi$ be its continuous piecewise-affine nodal interpolant on the contact edges. Under the additional obstacle hypotheses $\psi \le 0$ and convexity of $\psi$ in arclength on ${\mathrm{\Gamma }}_C$, define $K_h:=\{v_h\in V_h:v_h(a)\cdot n\ge I_h\psi (a)\text{ for every }a\in {\mathcal{N}}_h^C\}\mathrm{.}$ The nodal inequalities are exactly the plan's vertex constraints. The well-definedness and conformity claims are proved in The convex-obstacle nodal set is conforming ↗.