EIGENVALUES FOR THE SEMI-CIRCULANT PRECONDITIONING OF ELLIPTIC OPERATORS WITH THE VARIABLE COEFFICIENTS

Title & Authors
EIGENVALUES FOR THE SEMI-CIRCULANT PRECONDITIONING OF ELLIPTIC OPERATORS WITH THE VARIABLE COEFFICIENTS
Kim, Hoi-Sub; Kim, Sang-Dong; Lee, Yong-Hun;

Abstract
We investigate the eigenvalues of the semi-circulant preconditioned matrix for the finite difference scheme corresponding to the second-order elliptic operator with the variable coefficients given by $\small{L_vu\;:=-{\Delta}u+a(x,\;y)u_x+b(x,\;y)u_y+d(x,\;y)u}$, where a and b are continuously differentiable functions and d is a positive bounded function. The semi-circulant preconditioning operator $\small{L_cu}$ is constructed by using the leading term of $\small{L_vu}$ plus the constant reaction term such that $\small{L_cu\;:=-{\Delta}u+d_cu}$. Using the field of values arguments, we show that the eigenvalues of the preconditioned matrix are clustered at some number. Some numerical evidences are also provided.
Keywords
semi-circulant preconditioning;central finite difference;field of values;
Language
English
Cited by
References
1.
Z.-Z. Bai, O. Axelsson, and S.-X. Qiu, A class of nested iteration schemes for linear systems with a coefficient matrix with a dominant positive definite symmetric part, Numer. Algorithms 35 (2004), no. 2-4, 351-37

2.
Z.-Z. Bai, G. H. Golub and M. K. Ng, Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, SIAM J. Matrix Anal. Appl. 24 (2003), no. 3, 603-626

3.
Z.-Z. Bai, G. H. Golub, L.-Z. Lu, and J.-F. Yin, Block triangular and skew-Hermitian splitting methods for positive-definite linear systems, SIAM J. Sci. Comput. 26 (2005), no. 3, 844-863

4.
D. Bertaccini, G. H. Golub, S. Serra Capizzano, and C. T. Possio, Preconditioned HSS methods for the solution of non-Hermitian positive definite linear systems and applications to the discrete convection-diffusion equation, Numer. Math. 99 (2005), no. 3, 441-484

5.
R. Chan and T. Chan, Circulant preconditioners for elliptic problems, J. Numer. Linear Algebra Appl. 1 (1992), no. 1, 77-101

6.
R. Chan and X. Jin, A family of block preconditioners for block systems, SIAM J. Sci. Statist. Comput. 13 (1992), no. 5, 1218-1235

7.
R. Chan and M. K. Ng, Conjugate gradient methods for Toeplitz systems, SIAM Rev. 38 (1996), no. 3, 427-482

8.
W. M. Cheung and M. K. Ng, Block-circulant preconditioners for systems arising from discretization of the three-dimensional convection-diffusion equation, Proceedings of the 9th International Congress on Computational and Applied Mathematics (Leuven, 2000), J. Comput. Appl. Math. 140 (2002), no. 1-2, 143-158

9.
H. C. Elman and G. H. Golub, Iterative methods for cyclically reduced nonselfadjoint linear systems, Math. Comp. 54 (1990), no. 190, 671-700

10.
H. C. Elman and G. H. Golub, Iterative methods for cyclically reduced nonselfadjoint linear systems. II., Math. Comp. 56 (1991), no. 193, 215-242

11.
G. H. Golub, C. Grief, and J. M. Varah, Block orderings for tensor-product grids in two and three dimensions, Numer. Algorithms 30 (2002), no. 2, 93-11l

12.
G. H. Golub and D. Vanderstraeten, On the preconditioning of matrices with skewsymmetric splittings, Mathematical journey through analysis, matrix theory and scientific computation (Kent, OH, 1999), Numer. Algorithms 25 (2000), no. 1-4, 223-239

13.
K. E. Gustafson and D. K. M. Roo, Numerical range, The field of values of linear operators and matrices. Universitext. Springer-Verlag, New York, 1997

14.
L. Hemmingsson, A semi-circulant preconditioner for the convection-diffusion equation, Numer. Math. 81 (1998), no. 2, 211-248

15.
S. D. Kim and S. V. Parter, Semicirculant preconditioning of elliptic operators, SIAM J. Numer. Anal. 41 (2003), no. 2, 767-795

16.
A. R. Mitchell and D. F. Griffiths, The finite difference method in partial differential equations, A Wiley-Interscience Publication. John Wiley & Sons, Ltd., Chichester, 1980

17.
T. A. Manteuffel and J. Otto, Optimal equivalent preconditioners, SIAM J. Numer. Anal. 30 (1993), no. 3, 790-812

18.
T. A. Manteuffel and S. V. Parter, Preconditioning and boundary conditions, SIAM J. Numer. Anal. 27 (1990), no. 3, 656-694

19.
M. K. Ng, Circulant and skew-circulant splitting methods for Toeplitz systems, Proceedings of the 6th Japan-China Joint Seminar on Numerical Mathematics (Tsukuba, 2002). J. Comput. Appl. Math. 159 (2003), no. 1, 101-108