UBC Theses and Dissertations
Analysis and preconditioning of double saddle-point systems Bradley, Susanne
This thesis deals with the mathematical analysis and numerical solution of double saddle-point systems. We derive bounds on the eigenvalues of a generic form of double saddle-point matrices with a positive definite leading block. The bounds are expressed in terms of extremal eigenvalues and singular values of the associated block matrices. Inertia and algebraic multiplicity of eigenvalues are considered as well. The analysis includes bounds for preconditioned matrices based on block diagonal preconditioners using Schur complements, and it is shown that in this case the eigenvalues are clustered within a few intervals bounded away from zero. Analysis for approximations of Schur complements is included. Some numerical observations validate our analytical findings. We also derive bounds on the eigenvalues of (classical) saddle-point matrices with singular leading blocks. The technique of proof is based on augmentation. Our bounds depend on the principal angles between the ranges or kernels of the matrix blocks. We use these analyses to derive a preconditioner for saddle-point systems with singular leading blocks. Our preconditioning approach is based on augmenting the leading block and using Schur complements of the augmented system. We show that the resulting preconditioned operator has four distinct eigenvalues, and numerical experiments validate the effectiveness of our approach. We then extend the preconditioner for saddle-point systems with a singular leading block to deal with double saddle-point systems with a singular leading block. The preconditioner is based on augmenting the leading block by a null matrix of one of the off-diagonal blocks, and using the Schur complements of the augmented system.
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