These data are computed with the fully vectorial forward solver , where the tolerance for the BiCGSTAB
routine is set to [10.sup.-3].
One may note that the approximate inverse obtained per step of Algorithm (5) or (16) can also easily be taken into account as a preconditioner to reduce the ill-conditioning of a system and let the users apply iterative methods such as GMRES or BICGSTAB
in solving large scale sparse linear systems of algebraic equations efficiently.
Following the work , many other global methods have been developed, including, to name just a few of them, the global BiCG and global BiCGSTAB
methods [16, 17], the global Hessenberg and global CMRH (changing minimal residual method based on the Hessenberg process) methods  and their weighted variants , the skew-symmetric methods , and the global SCD method .
In the rest of this section, we apply our new iterative method as a robust technique to produce accurate preconditioners for accelerating modern iterative solvers such as GMRES or BiCGSTAB
for solving large scale sparse linear systems; see, for example, .
VAN DER VORST, Maintaining convergence properties of BiCGstab
methods infinite precision arithmetic, Numer.
Compared with the classical methods, which employ product-type methods such as GPBiCG or BiCGSTAB
as the iteration solver , the symmetry of the matrix enables preconditioned conjugate gradient (PCG) method to be employed to solve the interface problem of the domain decomposition system.
Preconditioned biconjugate gradient (PBiCG) BiCGstab
with a Cholesky preconditioning was used for the velocity and stress terms.
Among all the Krylov subspace methods for solving a linear system Ax = b with a nonsymmetric invertible coefficient matrix A, the stabilized bi-conjugate gradient algorithm (BiCGSTAB
) [van der Vorst 1992], the generalized minimal residual algorithm (GMRES) [Saad and Schultz 1986], and the quasi-minimal residual algorithm (QMR) [Freund and Nachtigal 1991] are considered the most robust [McQuain et al.
SADOK, A block version of BICGSTAB
for linear systems with multiple right-hand sides, Electron.
The total CPU time spent in the simulation has been 19,395 seconds using the BICGStab
iterative solver without any preconditioner.
Other conjugate gradient solvers, such as BiCGSTAB
, are also tested but PARDISO was found to be robust and more efficient in our work.
This study carefully explores the application of the Generalized Minimal Residual (GMRES) and the BiConjugate Gradient (BiCGstab
) methods to these problems, and recommends using the latter method with an incomplete LU preconditioning.