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Convergence estimates for product iterative methods with applications to domain decomposition. (English) Zbl 0754.65085

Let $V$ be a Hilbert space and ${V}^{\text{'}}$ its dual; let $\left\{{V}_{i}\right\}$, $\left\{{V}_{i}^{\text{'}}\right\}$ be sequences of closed subspaces of $V$, ${V}^{\text{'}}$ respectively, $A:V\to {V}^{\text{'}}$ a symmetric positive definite linear operator, and $f\in {V}^{\text{'}}$. To the equation $Au=f$ is attached the variational form $A\left(u,v\right)=〈f,v〉$, for all $v\in V$, respectively $〈{A}_{i}v,v〉=〈f,v〉$ for all $v\in {V}_{i}$, where ${A}_{i}$ is the restriction of $A$ to ${V}_{i}$, namely $〈{A}_{i}v,{\Phi }〉=A\left(v,{\Phi }\right)$, for all $v,{\Phi }\in {V}_{i}$.

Suppose that the linear operators ${R}_{i}:{V}_{i}^{\text{'}}\to {V}_{i}$ are given. If ${u}^{1}\in V$ is an approximation of the solution $u$ of the equation $Au=f$ or $A\left(u,v\right)=〈f,v〉$ for all $v\in V$, one puts ${y}_{0}={u}^{1}$ and ${y}_{i}={y}_{i-1}+{R}_{i}{Q}_{i}\left(f-A{y}_{i-1}\right)$, $i=1,\cdots ,J$ and defines ${u}^{\ell +1}={y}_{j}$, where ${Q}_{i}$ denotes the projection on ${V}_{j}^{\text{'}}$, namely $〈w-Qw,{\Phi }〉=0$, for all ${\Phi }\in {V}_{i}^{\text{'}}$.

The paper realizes a study of the convergence rate of ${u}^{\ell +1}$ to u, in terms of the product of the operators defined with respect to the number of subspaces ${V}_{i}$. If ${P}_{i}$ denotes the orthogonal projection in the subspace ${V}_{i}$ and ${T}_{i}={R}_{i}{A}_{i}{P}_{i}$ then $U-{u}^{\ell +1}=\left(I-{T}_{j}\right)\left(I-{T}_{j-1}\right)\cdots \left(I-{T}_{1}\right)\left(u-{u}^{\ell }\right)$ and in well specified hypotheses the above mentioned evaluations are obtained.

The applications to elliptic differential operators and a numerical model of finite element type for the plane case presented in the paper are suggestive.

##### MSC:
 65N30 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE) 65N55 Multigrid methods; domain decomposition (BVP of PDE) 65F10 Iterative methods for linear systems 65N15 Error bounds (BVP of PDE)