Preconditioned Conjugate Gradient Methods pp 44-57 | Cite as

# A class of preconditioned conjugate gradient methods applied to finite element equations

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## Abstract

It is well known that certain conditions have to be fulfilled to ensure existence of stable incomplete (IC) factorizations of a matrix.

Under additional conditions it is possible to construct *modified* incomplete (MIC) factorizations such that the order of convergence of the corresponding preconditioned conjugate gradient method is increased. For finite element discretizations of partial differential equations leading to matrices meeting these conditions, the number of iterations is of order *O*(*h*^{−1/2}), *h* → 0, compared to *O*(*h*^{−1}) for the standard incomplete (IC) factorizations. Here, *h* is a mesh-parameter.

Since particularly one of the conditions, the so called MIC-condition, sometimes is violated in applications, while the IC-conditions are fulfilled, this point of the theory will be stressed. The simplest method, the so called MIC(0)*-method, will be used as a model in this context.

In this survey, methods for satisfying the fundamental conditions, when they are not primarily fulfilled, will be discussed. Two such ideas are spectral equivalence and hierarchical finite element techniques. A number of applications, e g from plane strain linear elasticity problems, where these ideas are used, will be presented.

For certain model-problems, it is possible to construct modified incomplete factorizations based only on element-matrix factors, such that the actual increase of the rate of convergence is still obtained. This idea will be briefly discussed too.

## Key Words

Finite element equations preconditioned iterative methods conjugate gradients incomplete factorization## Preview

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## References

- [1]O. AXELSSON and I. GUSTAFSSON,
*Iterative Methods for the Solution of the Navier Equations of Elasticity*, Comput. Meths. Appl. Mech. Engrg. 15 (1978), 241–258.MathSciNetCrossRefzbMATHGoogle Scholar - [2]O. AXELSSON and I. GUSTAFSSON,
*On the use of Preconditioned Conjugate Gradient Methods for Red-Black Ordered Five-point Difference Schemes*, J. Comp. Physics 35 (1980), 284–289.MathSciNetCrossRefzbMATHGoogle Scholar - [3]O. AXELSSON and I. GUSTAFSSON,
*A Preconditioned Conjugate Gradient Method for Finite Element Equations, which is Stable for Rounding Errors*, Information Processing 80, ed: S. H. Lavington, North Holland Publishing Company (1980), 723–728.Google Scholar - [4]O. AXELSSON and I. GUSTAFSSON,
*Preconditioning and Two-Level Multigrid Methods of Arbitrary Degree of Approximation*, Math. Comp. 40 (1983), 219–242.MathSciNetCrossRefzbMATHGoogle Scholar - [5]O. AXELSSON and I. GUSTAFSSON,
*An Efficient Finite Element Method for Nonlinear Diffusion Problems*, Report 84.06 R, Department of Computer Sciences, Chalmers University of Technology, Göteborg, Sweden, 1984.zbMATHGoogle Scholar - [6]O. AXELSSON and P. S. VASSILEVSKI:
*Algebraic Multilevel Preconditioning Methods I*, Report 8811, Department of Mathematics, Catholic University, Nijmegen, Holland, 1988.zbMATHGoogle Scholar - [7]O. AXELSSON and G. LINDSKOG,
*A Recursive Two-Level Method for Boundary Value Problems Discretized by Quadratic Finite Elements*, Report 8, Numerical Analysis Group, Department of Computer Sciences, Chalmers University of Technology, Göteborg, Sweden, 1988.Google Scholar - [8]O. AXELSSON,
*On Iterative Solution of Elliptic Difference Equations on a Mesh-Connected Array of Processors*, International Journal of High Speed Computing 1 (1989), 165–183.CrossRefzbMATHGoogle Scholar - [9]O. AXELSSON, G. CAREY and G. LINDSKOG,
*On a Class of Preconditioned Iterative Methods for Parallel Computers*, Technical Report, Dept of Aerospace Eng. and Eng. Mech. The University of Texas at Austin, USA, 1989.zbMATHGoogle Scholar - [10]R. BLAHETA,
*An Incomplete Factorization Preconditioning Technique for Solving Linear Elasticity Problems*, submitted to the special issue of BIT on PCG methods, 1989.Google Scholar - [11]I. GUSTAFSSON,
*Stability and Rate of Convergence of Modified Incomplete Cholesky Factorization Methods, Report 79.02 R*, Department of Computer Sciences, Chalmers University of Technology, Göteborg, Sweden, 1979.Google Scholar - [12]I. GUSTAFSSON,
*Modified Incomplete Cholesky (MIC) Factorizations*, Preconditioning Methods-Theory and Applications, ed: D. Evans, Gordon and Breach Publishers, New York-London-Paris (1983), 265–293.Google Scholar - [13]I. GUSTAFSSON,
*A Preconditioned Iterative Method for the Solution of the Biharmonic Problem*, IMA Journal of Numerical Analysis 4 (1984), 55–67.MathSciNetCrossRefzbMATHGoogle Scholar - [14]I. GUSTAFSSON and G. LINDSKOG,
*A Preconditioning Technique based on Element Matrix Factorizations*, Comput. Meths. Appl. Mech. Engrg. 55 (1986), 201–220.MathSciNetCrossRefzbMATHGoogle Scholar - [15]M. JUNG, U. LANGER and U. SEMMLER,
*Preconditioned Conjugate Gradient Methods for Solving Linear Elasticity Finite Element Equations*, submitted to the special issue of BIT on PCG methods, 1989.Google Scholar - [16]E. F. KAASSCHIETER,
*A General Finite Element Preconditioning for the Conjugate Gradient Method*, submitted to the special issue of BIT on PCG methods, 1989.Google Scholar