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Zbl 0754.65085
Bramble, James H.; Pasciak, Joseph E.; Wang, Junping; Xu, Jinchao
Convergence estimates for product iterative methods with applications to domain decomposition.
(English)
[J] Math. Comput. 57, No.195, 1-21 (1991). ISSN 0025-5718; ISSN 1088-6842/e

Let $V$ be a Hilbert space and $V'$ its dual; let $\{V\sb i\}$, $\{V\sb i'\}$ 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 $\langle A\sb iv,v\rangle=\langle f,v\rangle$ for all $v\in V\sb i$, where $A\sb i$ is the restriction of $A$ to $V\sb i$, namely $\langle A\sb iv,\Phi\rangle=A(v,\Phi)$, for all $v,\Phi\in V\sb i$.\par Suppose that the linear operators $R\sb i: V\sb i'\to V\sb i$ are given. If $u\sp 1\in V$ 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 $y\sb 0=u\sp 1$ and $y\sb i=y\sb{i-1}+R\sb iQ\sb i(f-Ay\sb{i-1})$, $i=1,\dots,J$ and defines $u\sp{\ell+1}=y\sb j$, where $Q\sb i$ denotes the projection on $V\sb j'$, namely $\langle w-Qw,\Phi\rangle=0$, for all $\Phi\in V\sb i'$.\par The paper realizes a study of the convergence rate of $u\sp{\ell+1}$ to u, in terms of the product of the operators defined with respect to the number of subspaces $V\sb i$. If $P\sb i$ denotes the orthogonal projection in the subspace $V\sb i$ and $T\sb i=R\sb iA\sb iP\sb i$ then $U-u\sp{\ell+1}=(I-T\sb j)(I-T\sb{j-1})\dots(I-T\sb 1)(u-u\sp \ell)$ and in well specified hypotheses the above mentioned evaluations are obtained.\par The applications to elliptic differential operators and a numerical model of finite element type for the plane case presented in the paper are suggestive.
[T.Potra (Cluj-Napoca)]
MSC 2000:
*65N30 Finite numerical methods (BVP of PDE)
65N55 Multigrid methods; domain decomposition (BVP of PDE)
65F10 Iterative methods for linear systems
65N15 Error bounds (BVP of PDE)

Keywords: product iterative methods; algorithm; error-reducing operator; norm- reduction estimates; Schwarz algorithms; finite element; second-order elliptic equation; domain decomposition; Hilbert space

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