Mortar methods

Definition

Let $\Omega \subset \R^d$ be a bounded polygonal domain, divided into non-overlapping polygonal subdomains $\Omega_1, \dots, \Omega_N$. On each $\Omega_k$ let $X_k \subset H^1(\Omega_k)$ be a conforming finite element space on a mesh of $\Omega_k$. The meshes of neighboring subdomains need not match on their common boundary. Let $\Gamma_1, \dots, \Gamma_M$ be the interfaces, each a common side of two subdomains. For each $\Gamma_m$ choose one side as the slave side $s(m)$ and the other as the master side $\mu(m)$. Let $M_h(\Gamma_m)$ be a space of functions on $\Gamma_m$ (the mortar or multiplier space) built on the mesh of the slave side. Its dimension equals the number of slave degrees of freedom interior to $\Gamma_m$. The mortar finite element space is $$V_h = \Bigl\{ v : v|_{\Omega_k} \in X_k \text{ for all } k, \ \int_{\Gamma_m} \bigl( v|_{\Omega_{\mu(m)}} - v|_{\Omega_{s(m)}} \bigr) \psi \, ds = 0 \text{ for all } \psi \in M_h(\Gamma_m) \text{ and all } m \Bigr\},$$ where $v|_{\Omega_k}$ on $\Gamma_m$ means the trace of the restriction, together with the boundary conditions of the problem. A mortar method for a problem $a(u, v) = \ell(v)$ is the Galerkin method in $V_h$ with the broken form $a_h(u, v) = \sum_k a_{\Omega_k}(u, v)$.