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This paper studies the Craig variant of the Golub-Kahan bidiagonalizationalgorithm as an iterative solver for linear systems with saddle pointstructure. Such symmetric indefinite systems in 2x2 block form arise in manyapplications, but standard iterative solvers are often found to perform poorlyon them and robust preconditioners may not be available. Specifically, suchsystems arise in structural mechanics, when a semidefinite finite elementstiffness matrix is augmented with linear multi-point constraints via Lagrangemultipliers. Engineers often use such multi-point constraints to introduceboundary or coupling conditions into complex finite element models. The articlewill present a systematic convergence study of the Golub-Kahan algorithm for asequence of test problems of increasing complexity, including concretestructures enforced with pretension cables and the coupled finite element modelof a reactor containment building. When the systems are suitably transformedusing augmented Lagrangians on the semidefinite block and when the constraintequations are properly scaled, the Golub-Kahan algorithm is found to exhibitexcellent convergence that depends only weakly on the size of the model. Thenew algorithm is found to be robust in practical cases that are otherwiseconsidered to be difficult for iterative solvers.

An iterative generalized Golub-Kahan algorithm for problems in structural mechanics