Discrete harmonic extension and interface Schur complement

Use the interior/interface ordering of Scalar finite-element substructuring model. The boundary-trace spaces $W_i$, $W={\prod }_i{W}_i$, the conforming space $\widehat{W}$, the partially assembled space $\tilde{W}$, and the inclusion $R:\widehat{W}\to W$ are those of Primal interface spaces and jump convention. In particular, $R$ identifies the copies of a conforming interface trace. In subdomain $i$, write its stiffness matrix as

$K_i=\left(\begin{array}{cc}{K}_{II}^{(i)}& K_{I\mathrm{\Gamma }}^{(i)}\\ K_{\mathrm{\Gamma }I}^{(i)}& K_{\mathrm{\Gamma }\mathrm{\Gamma }}^{(i)}\end{array}\right)\mathrm{.}$

By Floating local kernels and positive interior blocks, $K_{II}^{(i)}$ is positive definite when nonempty; on the zero-dimensional interior space its unique endomorphism is invertible. For admissible boundary trace $g_i$, the discrete harmonic extension is

$H_i{g}_i:=\left(\begin{array}{c}-(K_{II}^{(i)}{)}^{-1}{K}_{I\mathrm{\Gamma }}^{(i)}{g}_i\\ g_i\end{array}\right)\mathrm{.}$

The local interface Schur complement is

$S_i:=K_{\mathrm{\Gamma }\mathrm{\Gamma }}^{(i)}-K_{\mathrm{\Gamma }I}^{(i)}(K_{II}^{(i)}{)}^{-1}{K}_{I\mathrm{\Gamma }}^{(i)}\mathrm{.}$

This is the Schur complement in Block LU factorisation and the associated Schur complement. The local Schur form on a trace is $g_i^T{S}_i{g}_i$. The block identity and the associated block LU factorization use An invertible leading block yields block LU through its Schur complement with leading block $K_{II}^{(i)}$. Assembling local Schur forms on $\widehat{W}$ gives $\widehat{S}=R^T({\mathrm{diag}}_i{S}_i)R$; restricting the same block-diagonal form to $\tilde{W}$ gives the partially assembled operator $\tilde{S}$.