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Preconditioning indefinite systems arising from mixed finite element discretization of second-order elliptic problems. (English) Zbl 0719.65024

Preconditioned conjugate gradient methods, Proc. Conf., Nijmegen/Neth. 1989, Lect. Notes Math. 1457, 28-43 (1990).
[For the entire collection see Zbl 0705.00024.]
The purpose of this paper is to study a question concerning some preconditioning techniques for solving indefinite systems of linear equations arising from mixed finite element discretizations of second order elliptic equations. The techniques are based on various approximations of the mass matrix, say, by simply lumping it to be diagonal or by constructing a diagonal matrix assembled of properly scaled lumped mass matrices.
The authors present here two possible alternatives for preconditioning. Main result: A precise upper bound of 1/2 for the spectral radius of the iteration matrix \(I-B^{-1}A\). They also prove that the eigenvalues of \(B^{-1}A\) are real and positive and that they are dominated by the eigenvalues of \(\tilde M^{-1}M\). Hence the generalized conjugate gradient method by O. Axelsson [Numer. Math. 51, 209-227 (1987; Zbl 0596.65014)] will have a convergence factor bounded by (\(\sqrt{\kappa}-1)/(\sqrt{\kappa}+1)\), where \(\kappa\) is the condition number of \(B^{-1}A\). Finally numerical experiments for some of the proposed iterative methods are presented.

MSC:

65F10 Iterative numerical methods for linear systems
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65F35 Numerical computation of matrix norms, conditioning, scaling
35J25 Boundary value problems for second-order elliptic equations