×
Kim, Sang Dong; Lee, Yong Hun

Analysis on eigenvalues for preconditioning cubic spline collocation method of elliptic equations. (English) Zbl 0981.65135

Linear Algebra Appl. 327, No. 1-3, 1-15 (2001).
The authors consider the cubic spline collocation methods for the Dirichlet-Neumann problem on the unit square for the equation \[ -\Delta u+ a_1(x, y)u_x+ a_2(x, y)u_y+ a(x, y)u= f \] and investigate efficient preconditioning techniques. They first analyze the eigenvalues of the resulting preconditioned matrix in the one-dimensional case. The two-dimensional analysis for eigenvalues, is basically developed from the one-dimensional argument.
Reviewer: Erwin Schechter (Kaiserslautern)

MSC:

65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
65F10 Iterative numerical methods for linear systems
35J25 Boundary value problems for second-order elliptic equations
65F35 Numerical computation of matrix norms, conditioning, scaling

Keywords:

elliptic equations; cubic spline collocation methods; Dirichlet-Neumann problem; preconditioning

Software:

Matlab; Spline Toolbox
Cite Review PDF
Full Text: DOI

References:

[1] Bialecki, B.; Fairweather, G.; Bennett, K. R., Fast direct solvers for piecewise Hermite bicubic orthogonal spline collocation equations, SIAM J. Numer. Anal., 29, 156-173 (1992) · Zbl 0752.65076
[2] C. de Boor, Spline Tool Box for Use with Matlab, The Mathworks Inc., 1996; C. de Boor, Spline Tool Box for Use with Matlab, The Mathworks Inc., 1996
[3] J.H. Bramble, J.E. Pasciak, Preconditioned iterative methods for nonself-adjoint or indefinite elliptic boundary value problems, in: H. Kardestuncer (Ed.), Unification of Finite Elements, Elsevier, Amsterdam, pp. 167-184; J.H. Bramble, J.E. Pasciak, Preconditioned iterative methods for nonself-adjoint or indefinite elliptic boundary value problems, in: H. Kardestuncer (Ed.), Unification of Finite Elements, Elsevier, Amsterdam, pp. 167-184 · Zbl 0544.65075
[4] Cerutti, J.; Parter, S. V., Collocation methods for parabolic partial differential equations in one-dimensional space, Numer. Math., 26, 227-254 (1974) · Zbl 0362.65094
[5] Christara, C. C.; Smith, B., Multigrid and mutilevel methods for quadratic spline collocation, BIT, 37, 781-803 (1997) · Zbl 0898.65081
[6] J. Douglas, T. Dupont, Collocation Methods for Parabolic Equations in a Single Space Variable, Lecture Notes in Mathematics, vol. 385, Springer, Berlin, 1974; J. Douglas, T. Dupont, Collocation Methods for Parabolic Equations in a Single Space Variable, Lecture Notes in Mathematics, vol. 385, Springer, Berlin, 1974 · Zbl 0279.65097
[7] Hajdjidimos, A.; Houstis, E. N.; Rice, J. R.; Vavalis, E., Analysis of iterative line spline collocation methods for elliptic partial differential equations, SIAM J. Matrix Anal. Appl., 21, 508-521 (1999) · Zbl 0947.65122
[8] Kim, H. O.; Kim, S. D.; Lee, Y. H., Finite difference preconditioning cubic spline collocation method of elliptic equations, Numer. Math., 77, 83-103 (1997) · Zbl 0879.65072
[9] Kim, S. D.; Parter, S. V., Preconditioning cubic spline collocation discretization of elliptic equations, Numer. Math., 72, 39-72 (1995) · Zbl 0844.65086
[10] S.V. Parter, Preconditioning legendre spectral collocation methods for elliptic problems: finite element operators, preprint; S.V. Parter, Preconditioning legendre spectral collocation methods for elliptic problems: finite element operators, preprint · Zbl 1004.65118
[11] Parter, S. V.; Wong, S.-P., Preconditioning second-order elliptic operators: condition numbers and the distribution of the singular values, J. Sci. Comput., 6, 1927-1957 (1991)
[12] Sun, W., Fast algorithms for high-order spline collocation systems, Numer. Math., 81, 143-160 (1998) · Zbl 0929.65103
[13] Sun, W., Spectral analysis of Hermite cubic spline collocation systems, SIAM J. Numer. Anal., 36, 1962-1975 (1999) · Zbl 0937.65110
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.