Iterative methods of gradient type in a subspace for a partial symmetric generalized eigenvalue problem. (English. Russian original) Zbl 0794.65037

Russ. Acad. Sci., Dokl., Math. 45, No. 2, 474-478 (1992); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 323, No. 6, 1020-1023 (1992).
Motivated by grid problems, the authors solve (1) \(Mu= \lambda Lu\) with \(M=M^*\), \(L=L^*>0\), \(u\in H\), a finite-dimensional Hilbert space, iteratively by (2) \(u^{k+1}= w^ k+ \tau_ k u^ k\), \(w^ k= B^{- 1} (Mu^ k- \alpha_ k Lu^ k)\) with preconditioner \(B=B^*>0\), \(\tau_ k\) numerical parameters. “Of gradient type” in the headline means that the Rayleigh quotient is replaced by a more general \(\alpha_ k\) such that \(\alpha_ k\to \lambda_ 1\) (the largest eigenvalue of (1)).
Earlier work of the authors used an orthogonal decomposition \(H= H_ 1\oplus H_ 2\) with \(\dim H_ 1\gg \dim H_ 2\) (e.g., in the domain decomposition method, \(H_ 1\) is the space of grid functions in the subdomains and \(H_ 2\) that of the functions of the separating boundary) and concerned iteration (2) in \(H_ 2\) for (1) with \(M=I\) with a \(B\) for which (2) is effective.
In the present paper this is extended to general \(M\) with two forms of \(B\) for which the method is not too cumbersome. A theorem on the rate of convergence is proved. A few data from numerical experiments for the Laplacian (5-point approximation) in an \(L\)-shaped domain are given.


65F15 Numerical computation of eigenvalues and eigenvectors of matrices
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation