High-order finite difference schemes and Toeplitz based preconditioners for elliptic problems

The coefficient function a(x) is assumed to be positive or with a finite number of zeros. The matrix Pn(a, m, k) is a Toeplitz based preconditioner constructed as D n (a, m, k)An(1, m, k)D 1/2 n (a, m, k), where Dn(a, m, k) is the suitably scaled diagonal part of An(a, m, k). The main result is the proof of the asymptotic clustering around unity of the eigenvalues of the preconditioned matrices. In addition, the “strength” of the cluster shows some interesting dependencies on the order k, on the regularity features of a(x) and on the presence of the zeros of a(x). The multidimensional case is analyzed in depth in a twin paper [38].

[1]  E. E. Tyrtyshnikov.  A unifying approach to some old and new theorems on distribution and clustering , 1996 .

[2]  J. Gillis,et al.  Matrix Iterative Analysis , 1961 .

[3]  O. Axelsson,et al.  On the rate of convergence of the preconditioned conjugate gradient method , 1986 .

[4]  S. Serra,et al.  Fast parallel solvers for elliptic problems , 1996 .

[5]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[6]  Stefano Serra,et al.  On the extreme eigenvalues of hermitian (block) toeplitz matrices , 1998 .

[7]  S. Holmgren,et al.  Iterative Solution Methods and Preconditioners for Block-Tridiagonal Systems of Equations , 1992, SIAM J. Matrix Anal. Appl..

[8]  Harold Widom,et al.  On the Singular Values of Toeplitz Matrices , 1989 .

[9]  Raymond H. Chan,et al.  Circulant preconditioners for elliptic problems , 1992 .

[10]  Stefano Serra Capizzano,et al.  Multigrid Methods for Symmetric Positive Definite Block Toeplitz Matrices with Nonnegative Generating Functions , 1996, SIAM J. Sci. Comput..

[11]  V. A. Barker,et al.  Finite element solution of boundary value problems , 1984 .

[12]  R. Noyé,et al.  Numerical Solutions of Partial Differential Equations , 1983 .

[13]  M. Schultz,et al.  Preconditioning by fast direct methods for nonself-adjoint nonseparable elliptic equations , 1986 .

[14]  U. Grenander,et al.  Toeplitz Forms And Their Applications , 1958 .

[15]  Dario Bini,et al.  SPECTRAL AND COMPUTATIONAL PROPERTIES OF BAND SYMMETRIC TOEPLITZ MATRICES , 1983 .

[16]  Seymour V. Parter.  On the extreme eigenvalues of truncated Toeplitz matrices , 1961 .

[17]  I. Gustafsson.  Stability and rate of convergence of modified incomplete Cholesky factorization methods , 1979 .

[18]  Stefano Serra Capizzano,et al.  The rate of convergence of Toeplitz based PCG methods for second order nonlinear boundary value problems , 1999, Numerische Mathematik.

[19]  Stefano Serra,et al.  Multigrid methods for toeplitz matrices , 1991 .

[20]  A. Böttcher,et al.  On the condition numbers of large semidefinite Toeplitz matrices , 1998 .

[21]  Stefano Serra,et al.  Optimal, quasi-optimal and superlinear band-Toeplitz preconditioners for asymptotically ill-conditioned positive definite Toeplitz systems , 1997 .

[22]  Stefano Serra Capizzano,et al.  Any Circulant-Like Preconditioner for Multilevel Matrices Is Not Superlinear , 2000, SIAM J. Matrix Anal. Appl..

[23]  Stefano Serra Capizzano,et al.  Spectral and structural analysis of high precision finite difference matrices for elliptic operators , 1999 .

[24]  Beatrice Meini,et al.  Effective Methods for Solving Banded Toeplitz Systems , 1999, SIAM J. Matrix Anal. Appl..

[25]  G. Golub,et al.  Block Preconditioning for the Conjugate Gradient Method , 1985 .

[26]  M. Krein.  On some new Banach algebras and Wiener-Levy type theorems for Fourier series and integrals , 1970 .

[27]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[28]  J. Meijerink,et al.  Guidelines for the usage of incomplete decompositions in solving sets of linear equations as they occur in practical problems , 1981 .

[29]  Stefano Serra,et al.  Preconditioning strategies for asymptotically ill-conditioned block Toeplitz systems , 1994 .

[30]  G. Szegő,et al.  On the Eigen-Values of Certain Hermitian Forms , 1953 .

[31]  Eugene E. Tyrtyshnikov,et al.  Circulant preconditioners with unbounded inverses , 1995 .

[32]  J. Meijerink,et al.  An iterative solution method for linear systems of which the coefficient matrix is a symmetric -matrix , 1977 .