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.

MSC:

 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