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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 \(V'\) 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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