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On the rate of convergence and complexity of normalized implicit preconditioning for solving finite difference equations in three space variables. (English) Zbl 1083.65102

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.

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
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