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Chen, Hao

University of British Columbia

Nonsymmetric saddle-point systems arise in numerous significant physical applications, including the Navier-Stokes equations and Stokes-Darcy coupling problems. The solution of these systems frequently employs preconditioned Generalized Minimal Residual (GMRES) method. To analyze convergence speed using preconditioners, field-of-values (FOV) analysis serves as a useful theoretical tool. While existing FOV analysis effectively characterizes the convergence of preconditioned GMRES when zero is not included in the field of values of the preconditioned matrix, it cannot be applied when zero is contained in the FOV. Recent theoretical advances by Crouzeix and Greenbaum [Spectral sets: numerical range and beyond. SIAM Journal on Matrix Analysis and Applications, 40(3):1087–1101, 2019] show that a convex region with a circular hole is a spectral set. We apply this result to nonsymmetric systems and derive sufficient conditions for convergence of preconditioned GMRES independent of the matrix dimensions. We apply our results to preconditioned nonsymmetric saddle-point systems and demonstrate applicability to previously unaddressed families of block preconditioners: specifically, a block diagonal preconditioner and a block upper triangular preconditioner for left preconditioning. A limitation of our theory is that the preconditioned matrix is required to have a small skew-symmetric part in norm. Consequently, our analysis may not be applicable, for example, to fluid flow problems characterized by a small viscosity number. To demonstrate the practical effectiveness of our theoretical analysis, we conduct numerical experiments on Navier-Stokes and Stokes-Darcy systems, which validate the performance of the proposed preconditioners and show that the iteration counts are independent of the matrix dimensions.

Text

2025-04-22

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/90701

Computer Science

University of British Columbia

UBCV

http://creativecommons.org/licenses/by-nc-nd/4.0/