A modified variant of HSS preconditioner for generalized saddle point problems

Abstract

Recently, Zhang [Numerical Linear Algebra with Applications, 2018: e2166] constructed an efficient variant of Hermitian and skew-Hermitian splitting (HSS) preconditioner for generalized saddle point problems, and gave the corresponding theoretical analysis and numerical experiment. In this paper, based on the efficient variant of the HSS (EVHSS) preconditioner, we relax the strong condition on the iterative parameter $\alpha$ , generalize the algorithms and further present the modified variant of Hermitian and skew-Hermitian splitting (HSS) (MVHSS) preconditioner for solving the generalized saddle point problem. Moreover, by similar theoretical analysis, we obtain that the MVHSS iterative method is convergent under weaker conditions. Finally, an example is given to verify the effectiveness of the proposed method.

MSC: 65F10; 65F15; 65F50

Introduction

Consider the following $2\times 2$ block saddle point problems

$$\mathcal{A}\left(\begin{array}{c}x\\ y\end{array}\right)\equiv \left(\begin{array}{cc}A& B^T\\ -B& C\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)=\left(\begin{array}{c}f\\ g\end{array}\right)=b,$$

(1)

where $A\in R^{n,n}$ is symmetric positive definite, $B\in R^{m,n},m\le n,$ is full rank, and $C\in R^{m,m}$ is symmetric and positive semidefinite. It appears in many different applications of scientific computing, such as constrained optimization, ⁴⁸ finite element method for solving Navier-Stokes equations,^26–28 and constrained least squares problem and generalized least squares problem^1,33,42,43 and so on; see^{9–23,31–47} and references therein.

In recent years, people have great interest in the saddle point problem of form (1), and a large number of stationary iterative methods have been proposed. For example, Santos et al. ³³ studied preconditioned iterative methods for solving the singular augmented system with $A=I$ . Golub et al. ²⁹ presented SOR-like algorithms for solving linear systems (1). Darvishi and Hessari ²⁵ studied SSOR method for solving the augmented systems. Bai et al., ² Bai and Wang, ³ Chen and Jiang, ²⁴ and Zheng et al. ⁴⁸ proposed the GSOR method, parameterized Uzawa (PU) and inexact parameterized Uzawa (PIU) methods for solving linear systems (1). Zhang and Lu ⁴⁴ showed the generalized symmetric SOR method for augmented systems. Peng and Li ³² studied unsymmetric block overrelaxation-type methods for saddle point. Bai and Yang, ⁴ Bai et al., ⁵ Bai, ⁶ Bai et al.,^7,8 Jiang and Cao, ³⁰ and Wang and Bai ³⁸ proposed split iteration methods, such as Hermitian and Skew-Hermitian splitting (HSS) iteration schemes and their pre-processing variables. Krylov subspace method, such as preprocessing conjugate gradient (PCG), preprocessing MINRES (PMINRES) and constrained preprocessing conjugate gradient (RPCG) iterative schemes, and related to Krylov subspace method, HSS, block diagonal method, block trigonometry, and constrained preprocessing techniques. Prof. Savin Treanta ³¹ propose and prove necessary and sufficient optimality conditions for multi-objective control problems involving multiple integrals. Professor Savin Treanta^34–36 studied the constraints of the constrained variational problem of the second-order Lagrange equation, the PDE optimal control problem and the first-order correction under the constraints of the relevant saddle point optimality criterion, as well as a new class of vector variational control problems.

More recently, by switching the position of the two separation matrices in the HSS preprocessor, Zhang ⁴⁵ presented an efficient variant of the Hermitian and skew-Hermitian splitting (HSS) preconditioner for generalized saddle point problems.Theoretical analysis and numerical examples show that under certain conditions, the corresponding iterative method converges to the unique solution of the generalized saddle point problem.

For large, sparse or structured matrices, iterative method is an attractive choice. In particular, Krylov subspace method applies the technology involving orthogonal projection to this form of subspace

$$\mathcal{K}(\mathcal{A},b)\equiv \mathrm{span}\{b,\mathcal{A}b,{\mathcal{A}}^{2}b,\dots ,{\mathcal{A}}^{n-1}b,\dots \}.$$

Conjugate gradient method, minimum residual method and generalized minimum residual method are commonly used Krylov subspace methods. CG method is used for symmetric positive definite matrix and MINRES method is used for symmetric possibly uncertain matrix. ³⁷

In this paper, based on the efficient variant of the HSS (EVHSS) preconditioner by Zhang, ⁴⁵ we added an acceleration factor and carried out theoretical analysis and numerical verification. Moreover, we obtain that the MVHSS iterative method is convergent under weaker conditions.

Modified variant of HSS (MVHSS) preconditioner

Recently, for the coefficient matrix of the augmented system (1), Zhang ⁴⁵ made the following splitting

$$\begin{array}{c}\mathcal{A}={\mathcal{P}}_{\mathrm{EVHSS}}-{\mathcal{R}}_{\mathrm{EVHSS}}\hfill \\ =\left(\begin{array}{cc}A\hfill & B^T(I+\frac{1}{\alpha }C)\hfill \\ \multicolumn{1}{c}{-B}& \alpha I+C\hfill \end{array}\right)-\left(\begin{array}{cc}0\hfill & \frac{1}{\alpha }{B}^{T}C\hfill \\ \multicolumn{1}{c}{0}& \alpha I\hfill \end{array}\right),\hfill \end{array}$$

(2)

where $\alpha >0$ are the constant number and $I$ is identity matrix (with appropriate dimension). Based on the iterative methods studied in Bai et al. ⁸ and Zhang ⁴⁵ we establish the modified variant of the Hermitian and skew-Hermitian splitting HSS (MVHSS) of the saddle point matrix $\mathcal{A}$ , which is as follows:

$$\begin{array}{c}\mathcal{A}={\mathcal{P}}_{\mathrm{MVHSS}}-{\mathcal{R}}_{\mathrm{MVHSS}}\hfill \\ =\left(\begin{array}{cc}A\hfill & B^T(\frac{\beta }{\alpha }I+\frac{1}{\alpha }C)\hfill \\ \multicolumn{1}{c}{-B}& \beta I+C\hfill \end{array}\right)-\left(\begin{array}{cc}0\hfill & (\frac{\beta }{\alpha }-1)B^T+\frac{1}{\alpha }{B}^{T}C\hfill \\ \multicolumn{1}{c}{0}& \beta I\hfill \end{array}\right),\hfill \end{array}$$

(3)

where $\alpha \ne 0,\beta \ne 0$ are two constant numbers. By this special splitting, the following modified variant of HSS splitting (MVHSS) method can be defined for saddle point problem (1):

Modified variant of the Hermitian and skew-Hermitian splitting (MVHSS) method

Given initial vectors $u^0\in R^{m+n}$ , and two relaxed parameters $\alpha \ne 0$ and $\beta \ne 0$ . For $k=0,1,2,...$ until the iteration sequence $\{u^k\}$ converges, compute

$$\begin{array}{c}\left(\begin{array}{cc}A\hfill & B^T(\frac{\beta }{\alpha }I+\frac{1}{\alpha }C)\hfill \\ \multicolumn{1}{c}{-B}& \beta I+C\hfill \end{array}\right)u^{k+1}\hfill \\ =\left(\begin{array}{cc}0\hfill & (\frac{\beta }{\alpha }-1)B^T+\frac{1}{\alpha }{B}^{T}C\hfill \\ \multicolumn{1}{c}{0}& \beta I\hfill \end{array}\right)u^k+\left(\begin{array}{c}\begin{array}{c}f\hfill \\ \multicolumn{1}{c}{g}\end{array}\hfill \end{array}\right),\hfill \end{array}$$

(4)

where $\alpha >0,\beta >0$ are two constant numbers. We can see that the iteration matrix of the MVHSS iteration is

$$\mathrm{\Gamma }={\left(\begin{array}{cc}A\hfill & B^T(\frac{\beta }{\alpha }I+\frac{1}{\alpha }C)\hfill \\ \multicolumn{1}{c}{-B}& \beta I+C\hfill \end{array}\right)}^{-1}\left(\begin{array}{cc}0\hfill & (\frac{\beta }{\alpha }-1)B^T+\frac{1}{\alpha }{B}^{T}C\hfill \\ \multicolumn{1}{c}{0}& \beta I\hfill \end{array}\right).$$

(5)

If the solution of the linear equations is approximated by the Generalized minimum residual method (GMRES) or its restarted variant Krylov subspace method, then

$${\mathcal{P}}_{\mathrm{MVHSS}}=\left(\begin{array}{cc}A\hfill & B^T(\frac{\beta }{\alpha }I+\frac{1}{\alpha }C)\hfill \\ \multicolumn{1}{c}{-B}& \beta I+C\hfill \end{array}\right)$$

(6)

can be served as a preconditioner. We call the preconditioner ${\mathcal{P}}_{\mathrm{MVHSS}}$ the MVHSS preconditioner for generalized saddle point matrix $\mathcal{A}$ .

In each iteration of mvhss iteration (4) or preprocessed Krylov subspace method, we need to solve a residual equation

$$\left(\begin{array}{cc}A\hfill & B^T(\frac{\beta }{\alpha }I+\frac{1}{\alpha }C)\hfill \\ \multicolumn{1}{c}{-B}& \beta I+C\hfill \end{array}\right)z=r$$

(7)

needs to be solved for a given vector $r$ at each step. Since the matrix ${\mathcal{P}}_{\mathrm{MVHSS}}$ has the following matrix factorization

$$\begin{array}{c}{\mathcal{P}}_{\mathrm{MVHSS}}=\frac{1}{\alpha }\left(\begin{array}{cc}I\hfill & \frac{1}{\alpha }{B}^T\hfill \\ \multicolumn{1}{c}{0}& I\hfill \end{array}\right)\left(\begin{array}{cc}A+\frac{1}{\alpha }{B}^{T}B\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& \alpha I\hfill \end{array}\right)\hfill \\ \left(\begin{array}{cc}I\hfill & 0\hfill \\ \multicolumn{1}{c}{-\frac{1}{\alpha }B}& I\hfill \end{array}\right)\left(\begin{array}{cc}\alpha I\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& \beta I+C\hfill \end{array}\right).\hfill \end{array}$$

(8)

Let $r=[r_1^T,r_2^T{]}^T$ and $z=[z_1^T,z_2^T{]}^T,$ where $r_1,z_1\in R^n$ and $r_2,z_2\in R^m.$ Then by (8), it can result in

$$\begin{array}{c}\left(\begin{array}{c}{z}_1\hfill \\ \multicolumn{1}{c}{z_2}\end{array}\right)=\alpha \left(\begin{array}{cc}\frac{1}{\alpha }I\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& {(\beta I+C)}^{-1}\hfill \end{array}\right)\left(\begin{array}{cc}I\hfill & 0\hfill \\ \multicolumn{1}{c}{\frac{1}{\alpha }B}& I\hfill \end{array}\right)\hfill \\ \left(\begin{array}{cc}{(A+\frac{1}{\alpha }{B}^{T}B)}^{-1}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& \frac{1}{\alpha }I\hfill \end{array}\right)\left(\begin{array}{cc}I\hfill & -\frac{1}{\alpha }{B}^T\hfill \\ \multicolumn{1}{c}{0}& I\hfill \end{array}\right)\left(\begin{array}{c}{r}_1\hfill \\ \multicolumn{1}{c}{r_2}\end{array}\right).\hfill \end{array}$$

(9)

Hence, analogous to Algorithm 2 in Zhang, ⁴⁵ we can derive the following algorithm about MVHSS iteration method.

• Algorithm 2.1. For given vector $r=[r_1^T,r_2^T{]}^T$ , and $z=[z_1^T,z_2^T{]}^T$ can be computed by (9) from the following steps:

• Step 1: Computed $t_1=r_1-\frac{1}{\alpha }{B}^T{r}_2;$

• Step 2: Solve $(A+\frac{1}{\alpha }{B}^{T}B)z_1=t_1;$

• Step 3: $(\beta I+C)z_2=B{z}_1+r_2$

• Remark 2.1. On MVHSS method, when $\beta =\alpha$ , the MVHSS method reduces to the EVHSS method. So, the MVHSS method is an extension of the existing iterative algorithm.

Covergence of MVHSS method

Now, we turn to the convergence of the MVHSS iterative solution of the generalized saddle point problem. We know that the iterative method (4) is convergent for every initial guess if and only if $\rho (\mathcal{T})<1,$ where $\rho (\mathcal{T})$ denotes the spectral radius of $\mathcal{T}$ . Based on EVHSS method, Zhang ⁴⁵ established the spectral properties of the iterative matrix and the preconditioned matrix ${\mathcal{P}}_{\mathrm{EVHSS}}^{-1}\mathcal{A}.$ In this section, we firstly obtain that the MVHSS iterative method is convergent under weaker conditions, then as a special case $\alpha =\beta$ , we relax the condition on the iteration parameter $\alpha$ of Theorems 3.1 and 3.2.

• Theorem 3.1. ⁴⁵ Suppose that the matrix $A\in R^{n,n}$ is symmetric positive definite, $B\in R^{m,n}$ is of full rank, $C\in R^{m,m}$ is symmetric positive semidefinite, and $\alpha$ is a positive constant. Let $u$ be a complex m-dimensional nonzero vector and $(\theta ,u)$ be an eigenpair of the matrix $(\alpha I+C{)}^{-1}(\alpha I+B{A}^{-1}{B}^T{)}^{-1}(B{A}^{-1}{B}^{T}C+{\alpha }^{2}I),$

$$\xi +\eta i=\frac{u^{*}B{A}^{-1}{B}^T\mathrm{Cu}}{u^{*}u}.$$

where $u^{*}$ denotes the conjugate transpose of the matrix $u\in R^n$ . If $\alpha$ and $\xi$ satisfy ${\alpha }^2+\xi >0,$ then it holds that

$$\rho (\mathcal{T})<1,$$

that is, the EVHSS iteration method converges to the unique solution of the generalized saddle point problem (1).

• Theorem 3.2. ⁴⁵ Suppose that the conditions in Theorem 3.1 are satisfied, and ${\mathcal{P}}_{\mathrm{EVHSS}}$ is defined as in (13) in Zhang. ⁴⁵ Then, the eigenvalues of the preconditioned matrix ${\mathcal{P}}_{\mathrm{EVHSS}}^{-1}\mathcal{A}$ satisfy the following:

• 1. ${\mathcal{P}}_{\mathrm{EVHSS}}^{-1}\mathcal{A}$ has an eigenvalue $1$ with multiplicity $n$ .

• 2. If $\alpha ,\beta$ and $\xi$ satisfy

$${\alpha }^2+\xi >0,$$

the real parts and the imaginary parts of the remaining nonunit eigenvalues ${\lambda }_i^{\prime }s$ s satisfy

$$0<\mathrm{Re}({\lambda }_i)<1$$

and

$$-\frac{1}{2}<\mathrm{Im}({\lambda }_i)<\frac{1}{2}.$$

• Theorem 3.3. Suppose that the matrix $A\in R^{n,n}$ is symmetric positive definite, $B\in R^{m,n}$ is of full rank, $C\in R^{m,m}$ is symmetric positive semidefinite, and $\alpha$ is a positive constant. Let $u$ be a complex m-dimensional nonzero vector and $(\theta ,u)$ be an eigenpair of the matrix $(\beta I+C{)}^{-1}(\alpha I+B{A}^{-1}{B}^T{)}^{-1}(B{A}^{-1}{B}^{T}C+{\alpha }^{2}I),$

$$\xi +\eta i=\frac{u^{*}B{A}^{-1}{B}^T\mathrm{Cu}}{u^{*}u}.$$

If $\alpha ,\beta$ , and $\xi$ satisfy

$${\alpha }^2+\xi >0,\mathrm{a}\mathrm{nd}\beta \ge \alpha >0$$

or

$${\alpha }^2+\xi <0,\mathrm{a}\mathrm{nd}\alpha \le \beta <0$$

then it holds that

$$\rho (\mathcal{T})<1,$$

that is, the MVHSS iteration method converges to the unique solution of the generalized saddle point problem (1).

Proof. The preconditioner ${\mathcal{P}}_{\mathrm{MVHSS}}$ has the following block triangular factorization:

$$\begin{array}{c}{\mathcal{P}}_{\mathrm{MVHSS}}=\frac{1}{\alpha }\left(\begin{array}{cc}A\hfill & B^T\hfill \\ \multicolumn{1}{c}{-B}& \alpha I\hfill \end{array}\right)\left(\begin{array}{cc}\alpha I\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& \beta I+C\hfill \end{array}\right)\hfill \\ =\frac{1}{\alpha }\left(\begin{array}{cc}I\hfill & \frac{1}{\alpha }{B}^T\hfill \\ \multicolumn{1}{c}{0}& I\hfill \end{array}\right)\left(\begin{array}{cc}A+\frac{1}{\alpha }{B}^{T}B\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& \alpha I\hfill \end{array}\right)\hfill \\ \left(\begin{array}{cc}I\hfill & 0\hfill \\ \multicolumn{1}{c}{-\frac{1}{\alpha }B}& I\hfill \end{array}\right)\left(\begin{array}{cc}\alpha I\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& \beta I+C\hfill \end{array}\right).\hfill \end{array}$$

(10)

By straightforward computation, we can obtain

$$\begin{array}{c}\mathcal{T}={\mathcal{P}}_{\mathrm{MVHSS}}^{-1}{\mathcal{R}}_{\mathrm{MVHSS}}\hfill \\ =\left(\begin{array}{cc}0\hfill & {(A+\frac{1}{\alpha }{B}^{T}B)}^{-1}(\frac{1}{\alpha }{B}^{T}C-B^T)\hfill \\ \multicolumn{1}{c}{0}& {(\beta I+C)}^{-1}[B{(A+\frac{1}{\alpha }{B}^{T}B)}^{-1}{B}^T(\frac{1}{\alpha }C-I)+\alpha I]\hfill \end{array}\right)\hfill \\ =\left(\begin{array}{cc}0\hfill & {(A+\frac{1}{\alpha }{B}^{T}B)}^{-1}{B}^T(\frac{1}{\alpha }C-I)\hfill \\ \multicolumn{1}{c}{0}& {(\beta I+C)}^{-1}{(\alpha I+B{A}^{-1}{B}^T)}^{-1}(B{A}^{-1}{B}^{T}C+{\alpha }^{2}I)\hfill \end{array}\right).\hfill \end{array}$$

(11)

Hence, $\mathcal{T}$ has an eigenvalue $0$ with multiplicity $n$ . Let $(\theta ,u)$ be an eigenpair of the matrix $(\beta I+C{)}^{-1}(\alpha I+B{A}^{-1}{B}^T{)}^{-1}(B{A}^{-1}{B}^{T}C+{\alpha }^{2}I).$ Then, it holds that

$$(B{A}^{-1}{B}^{T}C+{\alpha }^{2}I)u=\theta (\alpha I+B{A}^{-1}{B}^T)(\beta I+C)u.$$

(12)

Multiplying this equation by $\frac{u^{*}}{u^{*}u}$ from the left, we can rewrite the equation (12) as

$$\begin{array}{c}{\alpha }^2+\frac{U^{*}B{A}^{-1}{B}^T\mathrm{Cu}}{u^{*}u}=\theta [\alpha \beta +\frac{u^{*}(\beta B{A}^{-1}{B}^T+\alpha C)u}{u^{*}u}\hfill \\ +\frac{u^{*}B{A}^{-1}{B}^T\mathrm{Cu}}{u^{*}u}].\hfill \end{array}$$

(13)

Denote

$$\mu =\frac{u^{*}B{A}^{-1}{B}^{T}u}{u^{*}u},\gamma =\frac{u^{*}\mathrm{Cu}}{u^{*}u},\mathrm{a}\mathrm{nd}\xi +\eta i=\frac{u^{*}B{A}^{-1}{B}^T\mathrm{Cu}}{u^{*}u}.$$

Then, from equation (13) we have

$$\theta =\frac{{\alpha }^2+\xi +\eta i}{\alpha \beta +\beta \mu +\alpha \gamma +\xi +\eta i}$$

(14)

Consequently, the spectral radius of the iteration matrix $\mathcal{T}$ can be given by

$$\begin{array}{c}\rho (\mathcal{T})=max|\theta |=max\left|\frac{{\alpha }^2+\xi +\eta i}{\alpha \beta +\beta \mu +\alpha \gamma +\xi +\eta i}\right|\hfill \\ =max\sqrt{\frac{{({\alpha }^2+\xi )}^2+{\eta }^2}{{(\alpha \beta +\xi +\beta \mu +\alpha \gamma )}^2+{\eta }^2}}\hfill \end{array}$$

(15)

Because $A$ is symmetric positive definite, $B$ is full rank, and $C$ is symmetric positive semidefinite, it holds that $\mu >0,\gamma \ge 0$ .

Based on equation (15), if $\alpha ,\beta$ , and $\xi$ satisfy

$${\alpha }^2+\xi >0,\mathrm{a}\mathrm{nd}\beta \ge \alpha >0,$$

we have

$$\alpha \beta +\xi +\beta \mu +\alpha \gamma \ge {\alpha }^2+\xi +\beta \mu +\alpha \gamma >{\alpha }^2+\xi >0,$$

then, we can immediately derive from (15) that

$$\rho (\mathcal{T})<1.$$

Based on equation (15), if $\alpha ,\beta$ , and $\xi$ satisfy

$${\alpha }^2+\xi <0,\mathrm{a}\mathrm{nd}\alpha \le \beta <0,$$

we have

$$\alpha \beta +\xi +\beta \mu +\alpha \gamma \le {\alpha }^2+\xi +\beta \mu +\alpha \gamma <{\alpha }^2+\xi <0,$$

then, we can immediately derive from (15) that

$$\rho (\mathcal{T})<1.$$

that is, the MVHSS iterative converges to the unique solution of generalized saddle point problem (1) for any initial guess.

Now, we will establish the spectral properties of the preconditioned matrix ${\mathcal{P}}_{\mathrm{MVHSS}}^{-1}\mathcal{A}$ and analyze the bounds on the eigenvalues of the preconditioned matrix with the following theorem.

• Theorem 3.4. Suppose that the conditions in Theorem 3.2 are satisfied, and ${\mathcal{P}}_{\mathrm{MVHSS}}$ is defined as in (6). Then, the eigenvalues of the preconditioned matrix ${\mathcal{P}}_{\mathrm{MVHSS}}^{-1}\mathcal{A}$ satisfy the following:

• 1. ${\mathcal{P}}_{\mathrm{MVHSS}}^{-1}\mathcal{A}$ has an eigenvalue $1$ with multiplicity $n$ .

• 2. If $\alpha ,\beta$ and $\xi$ satisfy

$${\alpha }^2+\xi >0,\beta \ge \alpha >0$$

or

$${\alpha }^2+\xi <0,\alpha \le \beta <0,$$

the real parts and the imaginary parts of the remaining nonunit eigenvalues ${\lambda }_i^{″}s$ s satisfy

$$0<\mathrm{Re}({\lambda }_i)<1$$

and

$$-\frac{1}{2}<\mathrm{Im}({\lambda }_i)<\frac{1}{2}.$$

Proof. Since ${\mathcal{P}}_{\mathrm{MVHSS}}^{-1}\mathcal{A}=I-\mathrm{\Gamma },$ according to the expression of $\mathrm{\Gamma }$ in (11), the preconditioned matrix ${\mathcal{P}}_{\mathrm{MVHSS}}^{-1}\mathcal{A}$ has an eigenvalue $1$ with multiplicity $n$ , and the remaining eigenvalues ${\lambda }_i^{″}s(i=1,2,\cdots ,m)$ s satisfy

$$\begin{array}{c}{\lambda }_i=1-\theta =\frac{\alpha \beta +\beta \mu +\alpha \gamma -{\alpha }^2}{\alpha \beta +\beta \mu +\alpha \gamma +\xi +\eta i}\hfill \\ =\frac{\alpha \beta +\beta \mu +\alpha \gamma -{\alpha }^2}{{(\alpha \beta +\beta \mu +\alpha \gamma +\xi )}^2+{\eta }^2}\hfill \\ (\alpha \beta +\beta \mu +\alpha \gamma +\xi -\eta i)\hfill \end{array}$$

(16)

we find that ${\lambda }_i\to 0$ as $\alpha ,\beta \to 0$ or $\alpha ,\beta \to \infty .$ Moreover, the real parts of ${\lambda }_i$ satisfy

If ${\alpha }^2+\xi >0,\beta \ge \alpha >0,$ we have

$$\alpha \beta -{\alpha }^2\ge 0,$$

So, we can obtain

$$0<\alpha \beta +\beta \mu +\alpha \gamma -{\alpha }^2<\alpha \beta +\beta \mu +\alpha \gamma +\xi .$$

Then, we have

$$\begin{array}{c}\mathrm{Re}({\lambda }_i)=\frac{(\alpha \beta +\beta \mu +\alpha \gamma -{\alpha }^2)(\alpha \beta +\beta \mu +\alpha \gamma +\xi )}{{(\alpha \beta +\beta \mu +\alpha \gamma +\xi )}^2+{\eta }^2}\hfill \\ <\frac{{(\alpha \beta +\beta \mu +\alpha \gamma +\xi )}^2}{{(\alpha \beta +\beta \mu +\alpha \gamma +\xi )}^2+{\eta }^2}\hfill \\ <1.\hfill \end{array}$$

(17)

If ${\alpha }^2+\xi <0,\alpha \le \beta <0,$ we have

$$\alpha \beta -{\alpha }^2\le 0,$$

So, we can obtain

$$0>\alpha \beta +\beta \mu +\alpha \gamma -{\alpha }^2>\alpha \beta +\beta \mu +\alpha \gamma +\xi .$$

Then, we have

$$\begin{array}{c}\mathrm{Re}({\lambda }_i)=\frac{(\alpha \beta +\beta \mu +\alpha \gamma -{\alpha }^2)(\alpha \beta +\beta \mu +\alpha \gamma +\xi )}{{(\alpha \beta +\beta \mu +\alpha \gamma +\xi )}^2+{\eta }^2}\hfill \\ <\frac{{(\alpha \beta +\beta \mu +\alpha \gamma +\xi )}^2}{{(\alpha \beta +\beta \mu +\alpha \gamma +\xi )}^2+{\eta }^2}\hfill \\ <1.\hfill \end{array}$$

(18)

The imaginary parts of ${\lambda }_i$ satisfy

If ${\alpha }^2+\xi >0,\beta \ge \alpha >0,$ we have

$$\alpha \beta -{\alpha }^2\ge 0,$$

So, we can obtain

$$0<\alpha \beta +\beta \mu +\alpha \gamma -{\alpha }^2<\alpha \beta +\beta \mu +\alpha \gamma +\xi .$$

Then, we have

$$\begin{array}{c}|\mathrm{Im}({\lambda }_i)|=\left|\frac{-\eta (\alpha \beta +\beta \mu +\alpha \gamma -{\alpha }^2)}{{(\alpha \beta +\beta \mu +\alpha \gamma +\xi )}^2+{\eta }^2}\right|\hfill \\ =\frac{\eta (\alpha \beta +\beta \mu +\alpha \gamma -{\alpha }^2)}{{(\alpha \beta +\beta \mu +\alpha \gamma +\xi )}^2+{\eta }^2}\hfill \\ <\frac{\eta (\alpha \beta +\beta \mu +\alpha \gamma +\xi )}{{(\alpha \beta +\beta \mu +\alpha \gamma +\xi )}^2+{\eta }^2}\hfill \\ \le \frac{1}{2}\hfill \end{array}$$

(19)

If ${\alpha }^2+\xi <0,\alpha \le \beta <0,$ we have

$$\alpha \beta -{\alpha }^2\le 0,$$

So, we can obtain

$$0>\alpha \beta +\beta \mu +\alpha \gamma -{\alpha }^2>\alpha \beta +\beta \mu +\alpha \gamma +\xi .$$

Then, we have

$$\begin{array}{c}|\mathrm{Im}({\lambda }_i)|=\left|\frac{-\eta (\alpha \beta +\beta \mu +\alpha \gamma -{\alpha }^2)}{{(\alpha \beta +\beta \mu +\alpha \gamma +\xi )}^2+{\eta }^2}\right|\hfill \\ =\frac{-\eta (\alpha \beta +\beta \mu +\alpha \gamma -{\alpha }^2)}{{(\alpha \beta +\beta \mu +\alpha \gamma +\xi )}^2+{\eta }^2}\hfill \\ <\frac{-\eta (\alpha \beta +\beta \mu +\alpha \gamma +\xi )}{{(\alpha \beta +\beta \mu +\alpha \gamma +\xi )}^2+{\eta }^2}\hfill \\ \le \frac{1}{2}.\hfill \end{array}$$

(20)

As a special case of Theorems 3.3 and 3.4, we have the following results when $\beta =\alpha .$ Moreover, these results relax the condition on the iteration parameter $\alpha$ on Theorems 3.1 and 3.2.

• Theorem 3.5. Suppose that the matrix $A\in R^{n,n}$ is symmetric positive definite, $B\in R^{m,n}$ is of full rank, $C\in R^{m,m}$ is symmetric positive semidefinite, and $\alpha$ is a positive constant. Let $u$ be a complex m-dimensional nonzero vector and $(\theta ,u)$ be an eigenpair of the matrix $(\alpha I+C{)}^{-1}(\alpha I+B{A}^{-1}{B}^T{)}^{-1}(B{A}^{-1}{B}^{T}C+{\alpha }^{2}I),$

$$\xi +\eta i=\frac{u^{*}B{A}^{-1}{B}^T\mathrm{Cu}}{u^{*}u}.$$

If $\alpha ,\beta$ , and $\xi$ satisfy

$${\alpha }^2+\xi >0,\mathrm{a}\mathrm{nd}\beta =\alpha >0$$

or

$${\alpha }^2+\xi <0,\mathrm{a}\mathrm{nd}\alpha =\beta <0$$

then it holds that

$$\rho (\mathcal{T})<1,$$

In other words, the EVHSS iterative method converges to the unique solution of the generalized saddle point problem (1).

• Theorem 3.6. Suppose that the conditions in Theorem 3.2 are satisfied, and ${\mathcal{P}}_{\mathrm{EVHSS}}$ is defined as in (13) in Zhang. ⁴⁵ Then, the eigenvalues of the preconditioned matrix ${\mathcal{P}}_{\mathrm{MVHSS}}^{-1}\mathcal{A}$ satisfy the following:

• 1. ${\mathcal{P}}_{\mathrm{EVHSS}}^{-1}\mathcal{A}$ has an eigenvalue $1$ with multiplicity $n$ .

• 2. If $\alpha ,\beta$ , and $\xi$ satisfy

$${\alpha }^2+\xi >0,\beta =\alpha >0$$

or

$${\alpha }^2+\xi <0,\alpha =\beta <0,$$

the real parts and the imaginary parts of the remaining nonunit eigenvalues ${\lambda }_i^{\prime }s$ s satisfy

$$0<\mathrm{Re}({\lambda }_i)<1$$

and

$$-\frac{1}{2}<\mathrm{Im}({\lambda }_i)<\frac{1}{2}.$$

• Remark 3.1. On the one hand, MVHSS method is the generalization of EVHSS method. In addition, through the above theorems, we find that the conditions of Theorems 3.5 and 3.6 are weaker than Theorems 3.1 and 3.2 when $\alpha =\beta .$ On the other hand, according to different practical problems and different parameters, the problem will achieve faster convergence effect.

Numerical examples

In this section, we verify the above conclusions through numerical experiments. MATLAB 7.1 software was used to conduct numerical experiments, and the numerical experiment matrices were respectively generated based on the mixed two-dimensional time-harmonic Maxwell equations. In all of our runs, the relative residuals have been reduced by at least six orders of magnitude by the time we use zero initial guesses and stop iterating (i.e. when $\parallel b-\mathcal{A}{x}^k{\parallel }_2\le {10}^{-6}\parallel b{\parallel }_2$ ).

• Example 1. In this section, to further evaluate the effectiveness of the new block triangular preconditioned matrix ${\mathcal{P}}_{\mathrm{MVHSS}}^{-1}\mathcal{A}$ combined with Krylov subspace methods, we consider a numerical examples based on a two-dimensional time-harmonic Maxwell equations with constant coefficients: find the vector field $u$ and the multiplier $p$ such that vector field $u$ and the multiplier $p$ such that

$$\begin{array}{ccc}\nabla \times \nabla \times u+\nabla p\hfill & =\hfill & f\mathrm{i}\mathrm{n}\mathrm{\Omega },\hfill \\ \multicolumn{1}{c}{\nabla ·u}& =\hfill & 0\mathrm{i}\mathrm{n}\mathrm{\Omega },\hfill \\ \multicolumn{1}{c}{u\times n}& =\hfill & 0\mathrm{o}\mathrm{n}\partial \mathrm{\Omega },\hfill \\ \multicolumn{1}{c}{p}& =\hfill & 0\mathrm{o}\mathrm{n}\partial \mathrm{\Omega }.\hfill \end{array}$$

(21)

Here, $\mathrm{\Omega }\in R^3$ is simply connected polyhedron domain with a connected boundary $\partial \mathrm{\Omega }$ and $n$ denotes the outward unit normal on $\partial \mathrm{\Omega }$ . The datum $f$ is a given generic source. We know that finite element discretization using Nédélec elements of the first kind for the approximation of the vector field and standard nodal elements for the multiplier yields the above $2\times 2$ block saddle point problems.

For the simplicity, we take the generic source: $f=1$ and a finite element subdivision such as Figure 2 based on uniform grids of triangle elements. Three mesh sizes are considered: $h=\frac{\sqrt{2}}{8},\frac{\sqrt{2}}{12},\frac{\sqrt{2}}{18}.$ For example, $h=\frac{\sqrt{2}}{4}$ as shown in the Figure 1. The solution of the preprocessing system can be calculated accurately in each iteration. Table 1 shows the sparsity information of correlation matrices on different grids, where nz $(A)$ denote the nonzero elements of matrix $A$ .

Figure 1.

A uniform mesh with $h=\frac{\sqrt{2}}{4}$ .

Table 1.

Datasheet for different grids.

Grid	m	n	nz $(A)$	nz $(B)$	order of $\mathcal{A}$
$8\times 8$	176	49	820	462	225
$16\times 16$	736	225	3556	2190	961
$32\times 32$	3008	961	14,788	9486	3969
$64\times 64$	12,160	3969	60,292	39,438	16,129

Since the new preconditions have two parameters, we will test different values in numerical experiments. Numerical experiments show the spectrum of MVHSS preconditioned matrix ${\mathcal{P}}_{\mathrm{MVHSS}}^{-1}\mathcal{A}$ and EVHSS preconditioned matrix ${\mathcal{P}}_{\mathrm{EVHSS}}^{-1}\mathcal{A}$ when choosing different parameters, which coincides with theoretical analysis.

In Figures 2, 4, and 6 we display the eigenvalues of the preconditioned matrix ${\mathcal{P}}_{\mathrm{MVHSS}}^{-1}\mathcal{A}$ in the case of $h=\frac{\sqrt{2}}{8}$ , $h=\frac{\sqrt{2}}{12}$ , and $h=\frac{\sqrt{2}}{18}$ for different parameters. In Figures 3, 5, and 7 we show the eigenvalues of the preconditioned matrix ${\mathcal{P}}_{\mathrm{EVHSS}}^{-1}\mathcal{A}$ in the case of $h=\frac{\sqrt{2}}{8}$ , $h=\frac{\sqrt{2}}{12}$ , and $h=\frac{\sqrt{2}}{18}$ for different parameters. Figures $2~7$ 2–7 show that the distribution of eigenvalues of the preconditioned matrix confirms our above theoretical analysis. In Tables 2–4 we show iteration counts about preconditioned matrices ${\mathcal{P}}_{\mathrm{MVHSS}}^{-1}\mathcal{A}$ and ${\mathcal{P}}_{\mathrm{EVHSS}}^{-1}\mathcal{A}$ , when choosing different parameters and applying to BICGSTAB and GMRES Krylov subspace iterative methods on three meshes, where $I{t}_{\mathrm{BICGSTAB}({\mathcal{P}}_{\mathrm{MVHSS}}^{-1}\mathcal{A})}$ and $\mathrm{Re}{s}_{\mathrm{BICGSTAB}({\mathcal{P}}_{\mathrm{MVHSS}}^{-1}\mathcal{A})}$ are the iteration numbers and relative residual of the preconditioned matrices ${\mathcal{P}}_{\mathrm{MVHSS}}^{-1}\mathcal{A}$ when applying to BICGSTAB Krylov subspace iterative methods, respectively. $I{t}_{\mathrm{GMRES}({\mathcal{P}}_{\mathrm{MVHSS}}^{-1}\mathcal{A})}$ and $\mathrm{Re}{s}_{\mathrm{GMRES}({\mathcal{P}}_{\mathrm{MVHSS}}^{-1}\mathcal{A})}$ are the iteration numbers and relative residual of the preconditioned matrices ${\mathcal{P}}_{\mathrm{MVHSS}}^{-1}\mathcal{A}$ when applying to GMRES Krylov subspace iterative methods, respectively. $I{t}_{\mathrm{BICGSTAB}}$ and $\mathrm{Re}{s}_{\mathrm{BICGSTAB}}$ are the iteration numbers and relative residual of unpreconditioned matrices when applying to BICGSTAB Krylov subspace iterative methods, respectively. $I{t}_{\mathrm{BICGSTAB}({\mathcal{P}}_{\mathrm{EVHSS}}^{-1}\mathcal{A})}$ , $\mathrm{Re}{s}_{\mathrm{BICGSTAB}({\mathcal{P}}_{\mathrm{EVHSS}}^{-1}\mathcal{A})}$ , $I{t}_{\mathrm{GMRES}({\mathcal{P}}_{\mathrm{EVHSS}}^{-1}\mathcal{A})}$ , $\mathrm{Re}{s}_{\mathrm{GMRES}({\mathcal{P}}_{\mathrm{EVHSS}}^{-1}\mathcal{A})}$ are similar definitions.

• Remark 4.1. From the above figures and tables, we can see that MVHSS preconditioner ${\mathcal{P}}_{\mathrm{MVHSS}}$ has the same spectral clustering as the EVHSS preconditioner ${\mathcal{P}}_{\mathrm{EVHSS}}$ when choosing the optimal parameters.

• Remark 4.2. From Tables 2–4, we can see that the preconditioner ${\mathcal{P}}_{\mathrm{MVHSS}}$ and ${\mathcal{P}}_{\mathrm{EVHSS}}$ will improve the convergence of GERES and BICGSTAB iterative efficiently when they are applied to the preconditioned GMRES and BICGSTAB and to solve the two-dimensional time-harmonic Maxwell equations by choosing different parameters.

Figure 2.

Eigenvalue distribution of MVHSS preconditioned matrix ${\mathcal{P}}_{\mathrm{MVHSS}}^{-1}\mathcal{A}$ when $\alpha =0.2,\beta =0.4$ (the first), $\alpha =0.6,\beta =0.8$ (the second), $\alpha =0.9,\beta =1.4$ (the third), and $\alpha =1.2,\beta =1.6$ (the fourth), respectively. Here, $h=\frac{\sqrt{2}}{8}$ .

Figure 3.

Eigenvalue distribution of EVHSS preconditioned matrix ${\mathcal{P}}_{\mathrm{EVHSS}}^{-1}\mathcal{A}$ when $\alpha =0.2$ (the first), $\alpha =0.4$ (the second), $\alpha =0.8$ (the third), and $\alpha =1.2$ (the fourth), respectively. Here, $h=\frac{\sqrt{2}}{8}$ .

Figure 4.

Eigenvalue distribution of MVHSS preconditioned matrix ${\mathcal{P}}_{\mathrm{MVHSS}}^{-1}\mathcal{A}$ when $\alpha =0.2,\beta =0.4$ (the first), $\alpha =0.6,\beta =0.8$ (the second), $\alpha =0.9,\beta =1.4$ (the third), and $\alpha =1.2,\beta =1.6$ (the fourth), respectively. Here, $h=\frac{\sqrt{2}}{12}$ .

Figure 5.

Eigenvalue distribution of EVHSS preconditioned matrix ${\mathcal{P}}_{\mathrm{EVHSS}}^{-1}\mathcal{A}$ when $\alpha =0.2$ (the first), $\alpha =0.4$ (the second), $\alpha =0.8$ (the third), and $\alpha =1.2$ (the fourth), respectively. Here, $h=\frac{\sqrt{2}}{12}$ .

Figure 6.

Eigenvalue distribution of MVHSS preconditioned matrix ${\mathcal{P}}_{\mathrm{MVHSS}}^{-1}\mathcal{A}$ when $\alpha =0.2,\beta =0.4$ (the first), $\alpha =0.6,\beta =0.8$ (the second), $\alpha =0.9,\beta =1.4$ (the third), and $\alpha =1.2,\beta =1.6$ (the fourth), respectively. Here, $h=\frac{\sqrt{2}}{18}$ .

Figure 7.

Eigenvalue distribution of EVHSS preconditioned matrix ${\mathcal{P}}_{\mathrm{EVHSS}}^{-1}\mathcal{A}$ when $\alpha =0.2$ (the first), $\alpha =0.4$ (the second), $\alpha =0.8$ (the third), and $\alpha =1.2$ (the fourth), respectively. Here, $h=\frac{\sqrt{2}}{18}$ .

Table 2.

Iterative counting and relative residuals of preprocessing matrices ${\mathcal{P}}_{\mathrm{EVHSS}}^{-1}\mathcal{A}$ and ${\mathcal{P}}_{\mathrm{MVHSS}}^{-1}\mathcal{A}$ when choosing different parameters, where the number of iterations and relative residual of unpreconditioned BICGSTAB and GMRES are $381.5$ and $9.3594\times {10}^{-7}$ , $154(1)$ and $9.8351\times {10}^{-7}$ , respectively. Here, $h=\frac{\sqrt{2}}{8}$ denotes the size of the corresponding grid.

$\alpha$	$\beta$	$I{t}_{\mathrm{BICGSTAB}({\mathcal{P}}_{\mathrm{MVHSS}}^{-1}\mathcal{A})}$	$\mathrm{Re}{s}_{\mathrm{BICGSTAB}({\mathcal{P}}_{\mathrm{MVHSS}}^{-1}\mathcal{A})}$	$\alpha$	$I{t}_{\mathrm{BICGSTAB}({\mathcal{P}}_{\mathrm{EVHSS}}^{-1}\mathcal{A})}$	$\mathrm{Re}{s}_{\mathrm{BICGSTAB}({\mathcal{P}}_{\mathrm{EVHSS}}^{-1}\mathcal{A})}$
0.2	0.4	6	$4.2413\times {10}^{-7}$	0.2	6	$6.9379\times {10}^{-7}$
0.6	0.8	5	$5.5150\times {10}^{-7}$	0.4	5.5	$3.1353\times {10}^{-7}$
0.9	1.4	4.5	$9.9122\times {10}^{-7}$	0.8	5	$6.0483\times {10}^{-7}$
1.2	1.6	4.5	$8.7992\times {10}^{-7}$	1.2	5	$4.3066\times {10}^{-7}$
$\alpha$	$\beta$	$I{t}_{\mathrm{GMRES}({\mathcal{P}}_{\mathrm{MVHSS}}^{-1}\mathcal{A})}$	$\mathrm{Re}{s}_{\mathrm{GMRES}({\mathcal{P}}_{\mathrm{MVHSS}}^{-1}\mathcal{A})}$	$\alpha$	$I{t}_{\mathrm{GMRES}({\mathcal{P}}_{\mathrm{EVHSS}}^{-1}\mathcal{A})}$	$\mathrm{Re}{s}_{\mathrm{GMRES}({\mathcal{P}}_{\mathrm{EVHSS}}^{-1}\mathcal{A})}$
0.2	0.4	8 (1)	$2.0902\times {10}^{-7}$	0.2	8 (1)	$2.6884\times {10}^{-7}$
0.6	0.8	8 (1)	$3.6228\times {10}^{-7}$	0.4	8 (1)	$3.9882\times {10}^{-7}$
0.9	1.4	8 (1)	$2.7106\times {10}^{-7}$	0.8	8 (1)	$4.6599\times {10}^{-7}$
1.2	1.6	8 (1)	$2.7951\times {10}^{-7}$	1.2	8 (1)	$4.1991\times {10}^{-7}$

Table 3.

Iterative counting and relative residuals of preprocessing matrices ${\mathcal{P}}_{\mathrm{EVHSS}}^{-1}\mathcal{A}$ and ${\mathcal{P}}_{\mathrm{MVHSS}}^{-1}\mathcal{A}$ when choosing different parameters, where the number of iterations and relative residual of unpreconditioned BICGSTAB and GMRES are $381.5$ and $9.3594\times {10}^{-7}$ , $154(1)$ and $9.8351\times {10}^{-7}$ , respectively. Here, $h=\frac{\sqrt{2}}{12}$ denotes the size of the corresponding grid.

$\alpha$	$\beta$	$I{t}_{\mathrm{BICGSTAB}({\mathcal{P}}_{\mathrm{MVHSS}}^{-1}\mathcal{A})}$	$\mathrm{Re}{s}_{\mathrm{BICGSTAB}({\mathcal{P}}_{\mathrm{MVHSS}}^{-1}\mathcal{A})}$	$\alpha$	$I{t}_{\mathrm{BICGSTAB}({\mathcal{P}}_{\mathrm{EVHSS}}^{-1}\mathcal{A})}$	$\mathrm{Re}{s}_{\mathrm{BICGSTAB}({\mathcal{P}}_{\mathrm{EVHSS}}^{-1}\mathcal{A})}$
0.2	0.4	7.5	$3.2060\times {10}^{-7}$	0.2	8	$5.8212\times {10}^{-7}$
0.6	0.8	6	$7.7824\times {10}^{-7}$	0.4	6.5	$9.4471\times {10}^{-7}$
0.9	1.4	5.5	$4.6478\times {10}^{-7}$	0.8	6	$9.8843\times {10}^{-7}$
1.2	1.6	5.5	$3.8479\times {10}^{-7}$	1.2	5.5	$8.2576\times {10}^{-7}$
$\alpha$	$\beta$	$I{t}_{\mathrm{GMRES}({\mathcal{P}}_{\mathrm{MVHSS}}^{-1}\mathcal{A})}$	$\mathrm{Re}{s}_{\mathrm{GMRES}({\mathcal{P}}_{\mathrm{MVHSS}}^{-1}\mathcal{A})}$	$\alpha$	$I{t}_{\mathrm{GMRES}({\mathcal{P}}_{\mathrm{EVHSS}}^{-1}\mathcal{A})}$	$\mathrm{Re}{s}_{\mathrm{GMRES}({\mathcal{P}}_{\mathrm{EVHSS}}^{-1}\mathcal{A})}$
0.2	0.4	10 (1)	$9.7515\times {10}^{-7}$	0.2	12 (1)	$2.5747\times {10}^{-7}$
0.6	0.8	10 (1)	$9.6474\times {10}^{-7}$	0.4	11(1)	$3.2985\times {10}^{-7}$
0.9	1.4	10 (1)	$9.7589\times {10}^{-7}$	0.8	11 (1)	$2.5509\times {10}^{-7}$
1.2	1.6	10 (1)	$4.1596\times {10}^{-7}$	1.2	10 (1)	$8.0774\times {10}^{-7}$

Table 4.

Iterative counting and relative residuals of preprocessing matrices ${\mathcal{P}}_{\mathrm{EVHSS}}^{-1}\mathcal{A}$ and ${\mathcal{P}}_{\mathrm{MVHSS}}^{-1}\mathcal{A}$ when choosing different parameters, where the number of iterations and relative residual of unpreconditioned BICGSTAB and GMRES are $381.5$ and $9.3594\times {10}^{-7}$ , $154(1)$ and $9.8351\times {10}^{-7}$ , respectively. Here, $h=\frac{\sqrt{2}}{18}$ denotes the size of the corresponding grid.

$\alpha$	$\beta$	$I{t}_{\mathrm{BICGSTAB}({\mathcal{P}}_{\mathrm{MVHSS}}^{-1}\mathcal{A})}$	$\mathrm{Re}{s}_{\mathrm{BICGSTAB}({\mathcal{P}}_{\mathrm{MVHSS}}^{-1}\mathcal{A})}$	$\alpha$	$I{t}_{\mathrm{BICGSTAB}({\mathcal{P}}_{\mathrm{EVHSS}}^{-1}\mathcal{A})}$	$\mathrm{Re}{s}_{\mathrm{BICGSTAB}({\mathcal{P}}_{\mathrm{EVHSS}}^{-1}\mathcal{A})}$
0.2	0.4	8.5	$8.4785\times {10}^{-7}$	0.2	9	$9.9527\times {10}^{-7}$
0.6	0.8	7	$7.5396\times {10}^{-7}$	0.4	8	$6.6472\times {10}^{-7}$
0.9	1.4	6	$9.1395\times {10}^{-7}$	0.8	7	$8.3171\times {10}^{-7}$
1.2	1.6	6	$7.5320\times {10}^{-7}$	1.2	6.5	$5.6376\times {10}^{-7}$
$\alpha$	$\beta$	$I{t}_{\mathrm{GMRES}({\mathcal{P}}_{\mathrm{MVHSS}}^{-1}\mathcal{A})}$	$\mathrm{Re}{s}_{\mathrm{GMRES}({\mathcal{P}}_{\mathrm{MVHSS}}^{-1}\mathcal{A})}$	$\alpha$	$I{t}_{\mathrm{GMRES}({\mathcal{P}}_{\mathrm{EVHSS}}^{-1}\mathcal{A})}$	$\mathrm{Re}{s}_{\mathrm{GMRES}({\mathcal{P}}_{\mathrm{EVHSS}}^{-1}\mathcal{A})}$
0.2	0.4	14 (1)	$4.4195\times {10}^{-7}$	0.2	14 (1)	$9.7648\times {10}^{-7}$
0.6	0.8	13 (1)	$3.8744\times {10}^{-7}$	0.4	13 (1)	$8.7077\times {10}^{-7}$
0.9	1.4	11 (1)	$9.5295\times {10}^{-7}$	0.8	13 (1)	$4.8322\times {10}^{-7}$
1.2	1.6	11 (1)	$8.3118\times {10}^{-7}$	1.2	12 (1)	$5.6824\times {10}^{-7}$

Conclusions

In this paper, based on EVHSS preconditioner, we added an acceleration factor $\alpha$ , carried out theoretical analysis and numerical analysis. The purpose of this paper is to add an acceleration factor and then analyze the conditions of parameter convergence. In the actual numerical simulation, the convergence speed can be improved by selecting appropriate parameters.