Preconditioner

Numerical experiments

In this section we present the results of numerical experiments that illustrate the convergence behavior of the preconditioned GMRES [10], BICGSTAB [10], and FGM- RES [11] methods using the proposed preconditioner. All of the reported numerical results are run on a linux computer architecture. CPU times and iteration counts are reported in the rows ”CPU” and ”Iter” in the tables below. In the cases of the precon- ditioned GMRES and FGMRES methods, ”Iter”, namely, the number of steps of the preconditioned GMRES method or of the preconditioned FGMRES method. The norm of absolute residual vector denoted by RES. Here, ’RES’ is defined as RES = ǁb−AXǁ₂. In all the tables, a dagger ”†” indicates that the method has not converged in at most 200 iterations. Where P_α,Sˆ, P_T and P_D are given as follows:

_Aˆ
^Pα,S^{ˆ =} _Cˆ
Bˆ αS^ˆ
, P_T
Aˆ
= _Cˆ
^O and P = ^Aˆ
O . (28)

_Sˆ

D

O

_Sˆ
Where A^ˆ, B^ˆ, C^ˆ and S^ˆ are the approximation of A, B, C and Schur complement matrix
D − CA⁻¹B respectively. The parameter of the P_α,Sˆ preconditioner is chosen as to

implement the regularized preconditioner effeciently, we need to choose the parameters α appropriately since the analytic determination of the parameters which results in the fastest convergence of the preconditioned GMRES iteration appears to be quite a difficult problem. In Table 2, 3 and 4, to implement the regularized preconditioner effeciently, we need to choose the parameters α appropriately since the analytic determi- nation of the parameters which results in the fastest convergence of the preconditioned GMRES iteration appears to be quite a difficult problem. In the regularized precondi- tioner, the parameter α is taken as α = ǁB^ˆǁ₂ǁC^ˆǁ₂ /(ǁA^ˆǁ₂ǁS^ˆǁ₂), which balances the
matrices A^ˆ and C^ˆS^ˆ−1B^ˆ in the Euclidean norm. In all the numerical tests below, the
initial guess is taken to be the null vector. For the preconditioned GMRES, Flexible GMRES (FGMRES) and BICGSTAB methods, the iterations were stopped where :

ǁP⁻¹b − P⁻¹AX^(k)ǁ₂

ǁP⁻¹bǁ₂

_{< 10}−12


, (29)

∈
where P is one of the preconditioners P_α,Sˆ, P_T , or P_D, ǁ· ǁ₂ stands for the euclidien norm and X^(k) R^(n+m) denotes the current iterate. When using the inner precondi- tioned GMRES for solving the first system of Algorithm, the preconditioner used is a LU factorization, computed using MUMPS package [2, 3]. The inner relative residual norm less than (tol = 10⁻⁶).