Numerical behaviour of multigrid methods for symmetric Sinc–Galerkin systems
✍ Scribed by Michael K. Ng; Stefano Serra-Capizzano; Cristina Tablino-Possio
- Publisher
- John Wiley and Sons
- Year
- 2005
- Tongue
- English
- Weight
- 101 KB
- Volume
- 12
- Category
- Article
- ISSN
- 1070-5325
- DOI
- 10.1002/nla.418
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✦ Synopsis
Abstract
The symmetric Sinc–Galerkin method developed by Lund (Math. Comput. 1986; 47:571–588), when applied to second‐order self‐adjoint boundary value problems on d dimensional rectangular domains, gives rise to an N × N positive definite coefficient matrix which can be viewed as the sum of d Kronecker products among d − 1 real diagonal matrices and one symmetric Toeplitz‐plus‐diagonal matrix. Thus, the resulting coefficient matrix has a strong structure so that it can be advantageously used in solving the discrete system. The main contribution of this paper is to present and analyse a multigrid method for these Sinc–Galerkin systems. In particular, we show by numerical examples that the solution of a discrete symmetric Sinc–Galerkin system can be obtained in an optimal way only using O(N log N) arithmetic operations. Numerical examples concerning one‐ and two‐dimensional problems show that the multigrid method is practical and efficient for solving the above symmetric Sinc–Galerkin linear system. Copyright © 2004 John Wiley & Sons, Ltd.