Balancing domain decomposition

Statement

Let $\Omega \subset \R^d$, $d \in \set{2, 3}$, be a polygonal or polyhedral domain. Divide it into $N$ non-overlapping subdomains $\Omega_i$ of diameter about $H$, triangulated by a quasi-uniform mesh of size $h$. Let $\rho > 0$ be constant on each $\Omega_i$. Consider the finite element discretization of $-\nabla \cdot (\rho \nabla u) = f$ by continuous piecewise linear functions, with homogeneous Dirichlet conditions on part of $\partial \Omega$. Let $S = \sum_i R_i^T S_i R_i$ be the interface Schur complement, where $R_i$ restricts interface values to $\partial \Omega_i$ and $S_i$ is the local Schur complement of $\Omega_i$. Let $D_i$ be diagonal weight matrices with $\sum_i R_i^T D_i R_i = I$. Let $Z_i$ have columns spanning the null space of $S_i$ (the constants for a floating subdomain), and let $R_0^T$ have the columns $R_i^T D_i Z_i$. With $Q_0 = R_0^T (R_0 S R_0^T)^{-1} R_0$, the balancing domain decomposition preconditioner is $$M^{-1} = Q_0 + (I - Q_0 S) \Bigl( \sum_{i=1}^{N} R_i^T D_i S_i^{\dagger} D_i R_i \Bigr) (I - S Q_0),$$ where $S_i^\dagger$ is a generalized inverse of $S_i$. Choose the weights $D_i$ from the values of $\rho$ as in the standard theory. Then the condition number of $M^{-1} S$ satisfies $$\kappa(M^{-1} S) \le C \Bigl( 1 + \log \frac{H}{h} \Bigr)^2,$$ with $C$ independent of $h$, $H$, $N$ and the values of $\rho$.