Abstract
We investigate the solution of large linear systems of saddle point type with singular (1,1) block by preconditioned iterative methods and consider two parameterized block triangular preconditioners used with Krylov subspace methods which have the attractive property of improved eigenvalue clustering with increased ill-conditioning of the (1,1) block of the saddle point matrix, including the choice of the parameter. Meanwhile, we analyze the spectral characteristics of two preconditioners and give the optimal parameter in practice. Numerical experiments that validate the analysis are presented.
Highlights
We investigate the solution of large linear systems of saddle point type with singular (1, 1) block by preconditioned iterative methods and consider two parameterized block triangular preconditioners used with Krylov subspace methods which have the attractive property of improved eigenvalue clustering with increased ill-conditioning of the (1, 1) block of the saddle point matrix, including the choice of the parameter
We propose two new block triangular preconditioners
We have shown that in cases where the (1, 1) block has high nullity, convergence for each of the two preconditioned GMRES iterative methods is guaranteed to be almost immediate
Summary
ISRN Computational Mathematics systems with (1, 1) block that has a high nullity, Greif and Schotzau [25, 26] studied the application of the following block diagonal preconditioner used with the MINRES solver for the nonsymmetric saddle point systems (1): M [A. They have shown that if the nullity of A is m, which is the highest possible nullity, the preconditioned matrix M−1A has only two distinct eigenvalues 1 and −1.
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