Finite-dimensional Schwarz subspace-correction data

Work in exact real arithmetic. Let $n\ge 1$ and $V={\mathbb{R}}^n$, and let $A=A^{\mathsf{T}}$ satisfy $x^{\mathsf{T}}Ax>0$ for every nonzero $x$. Give $V$ the energy inner product $a(x,y)=x^{\mathsf{T}}Ay$ and norm $\mathrm{\parallel }x{\mathrm{\parallel }}_A^2=a(x,x)$ of The energy inner product $⟨u,v{⟩}_A=⟨Au,v⟩$ and energy norm $\mathrm{\parallel }u{\mathrm{\parallel }}_A$ for a Hermitian positive-definite matrix. An operator $S$ is $A$-self-adjoint when $a(Sx,y)=a(x,Sy)$ for all $x,y$; it is positive when $a(Sx,x)>0$ for every nonzero $x$. An $A$-orthogonal projection is an idempotent $A$-self-adjoint operator. Let $\mathcal{I}$ be a finite ordered nonempty set and, for each $i\in \mathcal{I}$, choose a full-column-rank prolongation $R_i\in {\mathbb{R}}^{n\times n_i}$ with $1\le n_i\le n$, and put $V_i=\mathrm{Ran}(R_i)$. A one-level method has $\mathcal{I}=\{1,\dots ,m\}$. A two-level method has $\mathcal{I}=\{0,1,\dots ,m\}$, where $R_0=P$ and $V_0$ is the designated coarse space. Restriction of a global residual is $R_i^{\mathsf{T}}$ and prolongation of a local coefficient vector is $R_i$. Define the local energy matrix $A_i=R_i^{\mathsf{T}}A{R}_i\mathrm{.}$ When the coarse index $0$ is present, $A_0=P^{\mathsf{T}}AP$ is the Galerkin coarse matrix of Galerkin coarse matrices and the exact coarse-correction error map. By The Galerkin coarse correction is an energy-orthogonal projector, it is SPD and the exact coarse solve is well-defined.

For each $i\in \mathcal{I}$, the exact local correction and additive preconditioner are ${\mathrm{\Pi }}_i=R_i{A}_i^{-1}{R}_i^{\mathsf{T}}A,B={\sum }_{i\in \mathcal{I}}{R}_i{A}_i^{-1}{R}_i^{\mathsf{T}}\mathrm{.}$ When the spaces span $V$, the exact additive Schwarz preconditioned operator is $BA={\sum }_{i\in \mathcal{I}}{\mathrm{\Pi }}_i$. For a fixed ordering of $\mathcal{I}$, a multiplicative sweep applies its corrections in that order; if $0$ is present, it is first. For an inexact additive method, replace each exact $A_i^{-1}$ by the inverse of an SPD matrix ${\tilde{A}}_i$ satisfying, for fixed $0<\alpha \le \beta$, $\alpha A_i⪯{\tilde{A}}_i⪯\beta A_i,\tilde{B}={\sum }_{i\in \mathcal{I}}{R}_i{\tilde{A}}_i^{-1}{R}_i^{\mathsf{T}}\mathrm{.}$ For symmetric matrices, $X⪯Y$ means $z^{\mathsf{T}}Xz\le z^{\mathsf{T}}Yz$ for every vector $z$. The order inequalities define uniform local spectral equivalence in this pair. The lemma Schwarz local matrices and exact solves are well defined ↗ proves the remaining local inverse and well-definedness claims.