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

Let $$V$$ be a Hilbert space and $$V'$$ its dual; let $$\{V_ i\}$$, $$\{V_ i'\}$$ be sequences of closed subspaces of $$V$$, $$V'$$ respectively, $$A: V\to V'$$ a symmetric positive definite linear operator, and $$f\in V'$$. To the equation $$Au=f$$ is attached the variational form $$A(u,v)=\langle f,v\rangle$$, for all $$v\in V$$, respectively $$\langle A_ iv,v\rangle=\langle f,v\rangle$$ for all $$v\in V_ i$$, where $$A_ i$$ is the restriction of $$A$$ to $$V_ i$$, namely $$\langle A_ iv,\Phi\rangle=A(v,\Phi)$$, for all $$v,\Phi\in V_ i$$.
Suppose that the linear operators $$R_ i: V_ i'\to V_ i$$ 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 $$y_ i=y_{i-1}+R_ iQ_ i(f-Ay_{i-1})$$, $$i=1,\dots,J$$ and defines $$u^{\ell+1}=y_ j$$, where $$Q_ i$$ denotes the projection on $$V_ j'$$, namely $$\langle w-Qw,\Phi\rangle=0$$, for all $$\Phi\in V_ i'$$.
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 $$T_ i=R_ iA_ iP_ i$$ then $$U-u^{\ell+1}=(I-T_ j)(I-T_{j-1})\dots(I-T_ 1)(u-u^ \ell)$$ 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 element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs 65F10 Iterative numerical methods for linear systems 65N15 Error bounds for boundary value problems involving PDEs
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