Primal constraints remove local floating kernels

From [F3], block multiplication gives $K_i{H}_i{g}_i=(0,S_i{g}_i{)}^T$ and therefore $(H_i{g}_i{)}^T{K}_i(H_i{g}_i)=g_i^T{S}_i{g}_i$. Thus a zero local Schur energy is precisely zero energy of its harmonic lift. In a local dual subproblem the primal values are prescribed, so a zero-energy local vector, or the harmonic lift of a zero-energy dual trace, is constant by [F1]. A primal corner value is among its prescribed values; because the dual correction has zero primal values, that constant is zero. The local dual quadratic form therefore has trivial kernel and, being symmetric positive semidefinite, is positive definite.

For $g=(g_i)\in \tilde{W}$, [F3] and step 1.1 give $g^T\tilde{S}g={\sum }_i{g}_i^T{S}_i{g}_i={\sum }_i(H_i{g}_i{)}^T{K}_i(H_i{g}_i)$. Each summand is a nonnegative local energy. If the partially assembled energy is zero, every summand is zero, so [F1] makes each local harmonic extension constant on its subdomain. Neighboring local constants agree at their shared primal corner. The adjacency graph is connected, so all constants agree; the global Dirichlet boundary makes the common value zero. Thus the partially assembled trace is zero, and its assembled energy matrix is positive definite.

The BDDC local dual solves use the positive local restrictions from step 1.1 and its coarse solve uses the positive partially assembled energy from step 2.1. The FETI-DP operator uses the same invertible partially assembled energy and the jump quotient of [F2]. Hence the exact local, coarse and dual systems named in the Statement are well-defined.