Falk error bound for nodal Signorini contact

Given: The continuous solution $u\in K$, a conforming P1 mesh family, the nodal feasible set $K_h\subset K$, and the restricted discrete problem on $V_h$.

Countable Choice is used by the continuous and finite-dimensional Stampacchia existence theorems. (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$))

The Axiom of Choice is used by the Lagrange nodal boundedness branch of the local P1 interpolation theorem. (The Axiom of Choice, Local interpolation estimates on shape-regular affine meshes)

The continuous solution satisfies $a(u,z-u)\ge F(z-u)$ for every $z\in K$. (Continuous frictionless elasticity contact is well posed)

$V_h$ 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)

The restrictions of $a$ and $F$ are bounded on $V_h$, and $a(v_h,v_h)\ge \alpha \mathrm{\parallel }{v}_h{\mathrm{\parallel }}_V^2$. (The isotropic contact elasticity form is well defined and bounded, Korn coercivity with a partial Dirichlet segment)

A bounded coercive bilinear form on a Hilbert space and a nonempty closed convex set have a unique variational-inequality solution under Countable Choice. (Stampacchia's variational inequality)

On the reference triangle ${\widehat{S}}_2$ with vertices $0,e_1,e_2$, the affine barycentric functions ${\widehat{\varphi }}_0=1-{\widehat{x}}_1-{\widehat{x}}_2$, ${\widehat{\varphi }}_1={\widehat{x}}_1$, and ${\widehat{\varphi }}_2={\widehat{x}}_2$ satisfy $p={\sum }_{i=0}^{2}p({\widehat{a}}_i){\widehat{\varphi }}_i$ for every $p\in {\mathbb{P}}_1$, so the three vertex evaluations are unisolvent and define the P1 Lagrange interpolant. The local interpolation theorem, applied with $d=2$, $p=2$, $k=1$, $r=\mathrm{\ell }=2$, and $m\in \{0,1\}$, gives $\mathrm{\mid }v-I_{K}v{\mathrm{\mid }}_{W^{m,2}(K)}\le C{h}_K^{2-m}\mathrm{\mid }v{\mathrm{\mid }}_{H^2(K)}$: its Axiom of Choice branch supplies boundedness of the vertex evaluations because $2>d\mathrm{/}p=1$. Apply the scalar estimate componentwise to vector fields, absorbing the fixed two-component norm factor into $C$; the local and global nodal interpolants agree because affine polynomials are determined by their three vertex values. (Local interpolation estimates on shape-regular affine meshes, Matching affine simplicial meshes and shape regularity, Reference finite elements and unisolvent degrees of freedom, The Axiom of Choice)

On a matching mesh, cell interiors are disjoint and the mesh fills the polygon; the finite union of straight edges has planar measure zero. (Matching affine simplicial meshes and shape regularity)