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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:VV ' a symmetric positive definite linear operator, and fV ' . To the equation Au=f is attached the variational form A(u,v)=f,v, for all vV, respectively A i v,v=f,v for all vV i , where A i is the restriction of A to V i , namely A i v,Φ=A(v,Φ), for all v,ΦV i .

Suppose that the linear operators R i :V i ' V i are given. If u 1 V is an approximation of the solution u of the equation Au=f or A(u,v)=f,v for all vV, one puts y 0 =u 1 and y i =y i-1 +R i Q i (f-Ay i-1 ), i=1,,J and defines u +1 =y j , where Q i denotes the projection on V j ' , namely w-Qw,Φ=0, for all ΦV i ' .

The paper realizes a study of the convergence rate of u +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 +1 =(I-T j )(I-T j-1 )(I-T 1 )(u-u ) 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.

65N30Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE)
65N55Multigrid methods; domain decomposition (BVP of PDE)
65F10Iterative methods for linear systems
65N15Error bounds (BVP of PDE)