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Asymmetric Hermitian and skew-Hermitian splitting methods for positive definite linear systems. (English) Zbl 1127.65019

Stationary iterative methods for a linear system of equations \(Ax = b\) can be constructed by an additive splitting of the matrix \(A\). Z.-Z. Bai, G. H. Golub and M. K. Ng [SIAM J. Matrix Anal. Appl. 24, No. 3, 603–626 (2003; Zbl 1036.65032)] considered the splitting \(A = H + S\) with \(H = (A+A^*)/2\) and \(S = (A-A^*)/2\), leading to the so called HSS iteration \[ (\alpha I + H)x^{(k+1/2)}= (\alpha I - S) x^{(k)} + b, \quad (\alpha I + S) x^{(k+1)} = (\alpha I - H) x^{(k+1/2)} + b, \] where \(\alpha\) is a fixed positive parameter. This paper proposes to allow for a different parameter in the second part of this iteration: \[ (\alpha I + H)x^{(k+1/2)} = (\alpha I - S) x^{(k)} + b, \quad (\beta I + S) x^{(k+1)} = (\beta I - H) x^{(k+1/2)} + b, \] where \(\alpha\) is nonnegative and \(\beta\) is positive. Bounds on \(\beta\) are given for which this modified iteration converges to \(x\). A bound on the spectral radius of the iteration matrix is given, depending on \(\alpha,\beta\) and the eigenvalues of \(H,M\). Optimal values of the parameters for a simplified bound are computed. It turns out that, for an arbitrary \(\alpha\), the choice \(\beta = \alpha\) can be far from optimal. On the other hand, numerical experiments for the finite difference discretization of a \(3D\) convection-diffusion equation suggest that little improvement over the HSS iteration is made when \(\alpha\) is chosen optimally. Finally, existing results on the influence of the inexact solution of the subsystems (e.g., by Krylov subspace methods) on the convergence are extended.

MSC:

65F10 Iterative numerical methods for linear systems
65N06 Finite difference methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations

Citations:

Zbl 1036.65032
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References:

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