A New Preconditioner with Two Variable Relaxation Parameters for Saddle Point Linear Systems with Highly Singular ( 1 , 1 ) Blocks

In this paper, we provide new preconditioner for saddle point linear systems with (1,1) blocks that have a high nullity. The preconditioner is block triangular diagonal with two variable relaxation paremeters and it is extension of results in [1] and [2]. Theoretical analysis shows that all eigenvalues of preconditioned matrix is strongly clustered. Finally, numerical tests confirm our analysis.


Introduction
Consider the following saddle point linear system , 0 where n is symmetric and positive semidefinite with nullity r, has full row rank, f R  and n g R  , u and p are unknown.Note that the assumption that A is nonsingular, i.e., the system (1) has a unique solution implies that null(F)/null(B) = 0, which we use in our analysis below.Saddle point linear systems of form (1) can arise, for example, from constraint optimization [3], mixed nite element formulation for the stokes problem [4], and discrete time harmonic Maxwell equations in mixed form [5].
There has many techniques for solving Saddle point linear systems of form (1), see [6] for a comprehensive survey.However, when F is singular, it cannot be inverted and the Schur complement does not exist.In this case, one possible way of dealing with system is by augmentation [7].Another way we can refer to [8] where Grief and SchÄotzau exploited a preconditioning technique for solving time-harmonic Maxwell equations in mixed form.
Recently, Rees and Grief [2] extend the work by Grief and SchÄotzau [8] to interior point methods for optimization problems.The preconditioner has attractive property of improved eigenvalue clustering with ill-conditioned the (1,1) block of saddle point systems.Based on the basic of above work, Huang etc. costructed two block triangular preconditioners for solving saddle point systems (1) [1].
In this paper we are devoted to give new block triangular preconditioner for solving saddle point systems of (1) with an ill-conditioned (1,1) blocks.The preconditioner is involving two parameters, and they are extension of recent work in Grief and SchÄotzau [8], Rees and Grief [2], and Cheng etc. [1].

New Preconditioner and Spectral Analysis
Rees and Grief [9] provided the following preconditioner for the symmetric saddle point systems (1) 1 , 0 where t is a scalar and is symmetric positive weight matrix.
Recently, Huang etc. [6] established the following preconditioners for the saddle point systems: In this section, we introduce the following preconditioner involving two parameters: for the saddle point systems (1), where 0   and 0   .We note that when the parameter where  are positive generalized eigenvalues of Let 1 i be a basis of the null space of F, be a basis of the null space of B, and From the second row we can obtain By substituting it into the first row we have   ; u W , then ( 5) is satisfied for any , and hence is an eigenvector of From which it follows that Proof: From Theorem 2.1 we obtain that the matrix with the algebraic multiplicities n and r, respectively.Taking inner product of ( 5) with u, we can obtain that the remaining m − r eigenvalues satisfy where ,   denotes the standard Euclidean inner product, u  null(F) and u  null(B).By (6), we have that the remaining m − r eigenvalues lie in interval When the parameters satisfy 1   has one eigenvalue which given by λ = 1 with algebraic multiplicity n + r.The remaining m − r eigenvalues lie in the interval (0, 1).
The following two-dimensional Model problem is considered: find u and p that satisfy in 0 in 0 on 0 on Here is a simply connected polyhedron domain with a connected boundary ∂Ω, and ~n denotes the outward unit normal on ∂Ω.The datum 2 R   f is a given source (not necessarily divergence free).Using the lowest order N'ed'elec elements of first kind [9,10] for the approximation of the vector field and standard nodal elements for multiplier yields the following saddle-point linear system .0 0 Experiments were done in a square domain (0 ≤ x ≤ 1; 0 ≤ y ≤ 1).And we set the right-hand side function so that the exact solution is given by In our numerical experiments the matrix W in the augmentation block preconditioner is taken as W = I.
We consider three meshes with different values of n and m in Table 1.Table 2 shows iteration counts for different η and meshes, applying BiCGSTAB of block-triangular preconditioner, and 1  1    .We ob-

1 ,
i a set of linearly independent vectors that complete null(F) null(B) to a basis of R n , Then the r vectors the n − m vectors , and the m − r vectors

has one eigenvalue which given by 1 
independent and form a subspace of n R of dimension n − m + r.Let   1 i i complete this set to a basis of Rn.It follows that  with algebraic multiplicity n + r.The remaining m − r eigenvalues satmultiplicities n and r, respectively.The remaining m − r eigenvalues lie in the interval

Table 2 . Iteration counts for different and meshes, using BiCGSTAB for solving the saddle point system with precon- ditioner   , the iteration was stopped once
[1]ve that for a fixed mesh, When η = 6, We spent the least iteration counts.In particulary, when η > 2, the iteration we spent are less than η = 2 (corresponding to M2[1]).It shows that preconditioner t M and ˆtM .