Dual P1 mortar multipliers for boundary contact

On a contact mesh with nodes $a_0,\dots ,a_N$ and edges $e_k=[a_k,a_{k+1}]$ of lengths $h_k>0$, let $W_h$ be the continuous piecewise-affine scalar trace space with nodal basis ${\varphi }_0,\dots ,{\varphi }_N$. On $e_k$, write the local endpoint basis as ${\varphi }_k^{e_k},{\varphi }_{k+1}^{e_k}$. Define broken piecewise-affine dual functions by ${\chi }_i{\mathrm{\mid }}_{e_k}=\{\begin{array}{ll}4{\varphi }_k^{e_k}-2{\varphi }_{k+1}^{e_k},& i=k,\\ -2{\varphi }_k^{e_k}+4{\varphi }_{k+1}^{e_k},& i=k+1,\\ 0,& i\mathrm{\notin }\{k,k+1\},\end{array}$ and set ${\mathrm{\Lambda }}_h:=\mathrm{span}\{{\chi }_i:0\le i\le N\}\subset L^2({\mathrm{\Gamma }}_C)$. The positive discrete multiplier cone is ${\mathrm{\Lambda }}_h^{+}:=\{{\lambda }_h={\sum }_{i=0}^N{\lambda }_i{\chi }_i:{\lambda }_i\ge 0\}\mathrm{.}$ The functions ${\chi }_i$ have support on the one or two contact edges incident to $a_i$ and can jump at contact nodes. For $v_h\in V_h$ and ${\lambda }_h\in {\mathrm{\Lambda }}_h$, define the exact mortar coupling $b_h(v_h,{\lambda }_h):={\int }_{{\mathrm{\Gamma }}_C}{\lambda }_h{T}_C{v}_h\cdot nds\mathrm{.}$ The cone ${\mathrm{\Lambda }}_h^{+}$ is coefficientwise; its elements need not be pointwise nonnegative. Its dual-cone and stability properties are proved in The dual mortar basis is biorthogonal and uniformly stable on the trace space ↗. This is a matching boundary mortar pair and makes no nonmatching-interface claim.