Gravvanis, George A.; Giannoutakis, Konstantinos M. On the rate of convergence and complexity of normalized implicit preconditioning for solving finite difference equations in three space variables. (English) Zbl 1083.65102 Int. J. Comput. Methods 1, No. 2, 367-386 (2004). Implicit preconditioning methods, based on approximate sparse factorization procedures, have been extensively used for solving sparse linear systems, which are derived from the discretization of partial differential equations, and are of central importance in computational science and engineering. The authors consider a class of problems defined by the following self-adjoint partial differential equation in three space variables: \[ \sum_{i = 1}^{N = 3} \frac{\partial } {\partial x_i }\left( \alpha _i (x)\frac{\partial u} {\partial x_i} \right) + P(x)u(x) + Q(x) = 0, \quad x \equiv (x_1 ,x_2 ,x_3) \in \Omega \] subject to the general boundary conditions: \( \alpha (x)u + \beta(x)\frac{\partial u} {\partial \eta} = \gamma (x)\), \(x \equiv (x_1 ,x_2 ,x_3) \in \partial\Omega,\) where \(\alpha _1(x) > 0\), \(\alpha _2 (x)> 0\), \(\alpha _3(x) > 0\), \(P(x)\leq 0\) and \(Q\) are sufficiently smooth functions on \(\Omega\). Assuming a column-wise ordering is used, the linear system derived from the finite difference discretization method applied to the partial differential equation in three space variables is \[ Au = s, \] where the coefficient matrix \(A\) is a nonsingular sparse symmetric positive definite diagonally dominant \((n \times n)\) matrix. In Section 2, the authors present modified forms of the normalized approximate factorization procedures of the coefficient matrix \(A \). In Section 3, normalized implicit preconditioned conjugate gradient-type methods are introduced. Theoretical results on the rate of convergence and estimates of the computational complexity are given in Section 4. Finally, the performance and applicability of the normalized implicit preconditioned methods is illustrated by solving characteristic elliptic boundary value problems and numerical results are given. Reviewer: Leonid B. Chubarov (Novosibirsk) Cited in 1 Document MSC: 65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs 65F05 Direct numerical methods for linear systems and matrix inversion 65F35 Numerical computation of matrix norms, conditioning, scaling 65F50 Computational methods for sparse matrices 65N06 Finite difference methods for boundary value problems involving PDEs 35J25 Boundary value problems for second-order elliptic equations Keywords:implicit preconditioning methods; approximate sparse factorization procedures; self-adjoint partial differential equation; rate of convergence; computational complexity; elliptic boundary value problems; finite difference discretization method; parse symmetric positive definite diagonally dominant matrix; conjugate gradient-type methods; numerical results PDFBibTeX XMLCite \textit{G. A. Gravvanis} and \textit{K. M. Giannoutakis}, Int. J. Comput. Methods 1, No. 2, 367--386 (2004; Zbl 1083.65102) Full Text: DOI References: [1] DOI: 10.1017/CBO9780511624100 · doi:10.1017/CBO9780511624100 [2] DOI: 10.1007/BF01389448 · Zbl 0564.65017 · doi:10.1007/BF01389448 [3] Dongarra J. J., Solving linear systems on vector and shared memory computers (1991) [4] DOI: 10.1016/0045-7825(80)90075-4 · Zbl 0447.65011 · doi:10.1016/0045-7825(80)90075-4 [5] Golub G. H., Matrix Computations (1996) · Zbl 0865.65009 [6] Gravvanis G. A., The Journal of Supercomputing [7] DOI: 10.1137/1.9781611970937 · doi:10.1137/1.9781611970937 [8] DOI: 10.1002/nla.1680010208 · Zbl 0837.65027 · doi:10.1002/nla.1680010208 [9] DOI: 10.1016/0024-3795(86)90227-2 · Zbl 0609.65070 · doi:10.1016/0024-3795(86)90227-2 [10] DOI: 10.1016/0771-050X(81)90011-5 · Zbl 0453.65014 · doi:10.1016/0771-050X(81)90011-5 [11] Lipitakis E. A., Mathematics and Computers in Simulation pp 189– [12] Lipitakis E. A., Special Issue on Computer Mathematics, Bulletin of the Greek Mathematical Society 33 pp 11– [13] DOI: 10.1007/BF01385754 · Zbl 0791.65016 · doi:10.1007/BF01385754 [14] Saad Y., Iterative Methods for Sparse Linear Systems (1996) · Zbl 1031.65047 [15] DOI: 10.1137/0910004 · Zbl 0666.65029 · doi:10.1137/0910004 [16] DOI: 10.1007/BF01389450 · Zbl 0596.65015 · doi:10.1007/BF01389450 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.