*(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(u,v)=\langle f,v\rangle $, for all $v\in V$, respectively $\langle {A}_{i}v,v\rangle =\langle f,v\rangle $ for all $v\in {V}_{i}$, where ${A}_{i}$ is the restriction of $A$ to ${V}_{i}$, namely $\langle {A}_{i}v,{\Phi}\rangle =A(v,{\Phi})$, 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(u,v)=\langle f,v\rangle $ for all $v\in V$, one puts ${y}_{0}={u}^{1}$ and ${y}_{i}={y}_{i-1}+{R}_{i}{Q}_{i}(f-A{y}_{i-1})$, $i=1,\cdots ,J$ and defines ${u}^{\ell +1}={y}_{j}$, where ${Q}_{i}$ denotes the projection on ${V}_{j}^{\text{'}}$, namely $\langle w-Qw,{\Phi}\rangle =0$, 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 $U-{u}^{\ell +1}=(I-{T}_{j})(I-{T}_{j-1})\cdots (I-{T}_{1})(u-{u}^{\ell})$ 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) |