Ordering Methods for Preconditioned Conjugate Gradient Methods Applied to Unstructured Grid Problems

It is well known that the ordering of the unknowns can have a significant effect on the convergence of preconditioned conjugate gradient (PCG) methods. There has been considerable experimental work on the effects of ordering for finite difference problems. In many cases, good results have been obtai...

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Bibliographic Details
Published in:SIAM journal on matrix analysis and applications 1992-07, Vol.13 (3), p.944-961
Main Authors: D’Azevedo, E. F., Forsyth, P. A., Tang, Wei-Pai
Format: Article
Language:English
Subjects:
540200 - Environment, Terrestrial- (1990-)
990200 - Mathematics & Computers
Applied mathematics
CALCULATION METHODS
Computer science
CONTAMINATION
CONVERGENCE
ENVIRONMENTAL SCIENCES
FACTORIZATION
FINITE DIFFERENCE METHOD
GROUND WATER
Groundwater
Groundwater pollution
HYDROGEN COMPOUNDS
ITERATIVE METHODS
MATHEMATICAL MODELS
MATHEMATICS AND COMPUTING
MESH GENERATION
Methods
NUMERICAL SOLUTION
OXYGEN COMPOUNDS
POLLUTION
Sparsity
WATER
WATER POLLUTION
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Summary:It is well known that the ordering of the unknowns can have a significant effect on the convergence of preconditioned conjugate gradient (PCG) methods. There has been considerable experimental work on the effects of ordering for finite difference problems. In many cases, good results have been obtained with preconditioners based on diagonal, spiral, red / black reduced system orderings, or some others. The reduced system approach generally gives rapid convergence. There has been comparatively less work on the effect of ordering for finite element problems on unstructured meshes. In this paper, an ordering technique for unstructured grid problems is developed. At any stage of the partial elimination, the next pivot node is selected so as to minimize the norm of the discarded fill matrix. Numerical results are given for model problems and for problems arising in groundwater contamination. Computations are reported for two-dimensional triangular grids, and for three-dimensional tetrahedral grids. The examples show that ordering is important even if a reduced system (based on a generalized red/black ordering) method is used.
ISSN:0895-4798
1095-7162
DOI:10.1137/0613057