Convergence Analysis of the Preconditioned Group Splitting Methods in Boundary Value Problems

Abstract

The construction of a specific splitting-type preconditioner in block formulation applied to a class of group relaxation iterative methods derived from the centred and rotated (skewed) finite difference approximations has been shown to improve the convergence rates of these methods. In this paper, we present some theoretical convergence analysis on this preconditioner specifically applied to the linear systems resulted from these group iterative schemes in solving an elliptic boundary value problem. We will theoretically show the relationship between the spectral radiuses of the iteration matrices of the preconditioned methods which affects the rate of convergence of these methods. We will also show that the spectral radius of the preconditioned matrices is smaller than that of their unpreconditioned counterparts if the relaxation parameter is in a certain optimum range. Numerical experiments will also be presented to confirm the agreement between the theoretical and the experimental results.

1. Introduction

Consider the finite difference discretisation schemes for solving the following boundary value problem which is the two-dimensional Poisson equation with Dirichlet boundary conditions: 𝑢𝑢_𝑥𝑥(𝑥+,𝑦𝑢)_𝑦𝑦==𝑔𝑓(𝑥(,𝑥𝑦,)𝑦,),(𝑥(,𝑥𝑦,)𝑦∈)∈𝜕ΩΩ.,(1.1) Here, Ω is a continuous unit square solution domain with boundary 𝜕Ω. This equation plays a very important role in the modelers of fluid flow phenomena and heat conduction problems. Let Ω be discretised uniformly in both 𝑥 and 𝑦 directions with a mesh size ℎ=1/𝑛, where 𝑛 is an integer. The simplest finite difference approximation of the Laplacian is 𝑢_𝑖+1,𝑗−2𝑢_𝑖𝑗+𝑢_{𝑖−1,𝑗}ℎ²+𝑢^𝑖,𝑗+1−2𝑢_𝑖𝑗+𝑢_{𝑖,𝑗−1}ℎ²−112ℎ²𝜕⁴𝑢𝜕𝑥⁴+𝜕⁴𝑢𝜕𝑦⁴−𝑂ℎ⁴=𝑓_𝑖𝑗.(1.2) Here, 𝑢_𝑖𝑗=𝑢(𝑥_𝑖,𝑦_𝑗). Another approximation to (1.1) can be derived from the rotated five-point finite-difference approximation to give [1] 𝑢_{𝑖+1,𝑗+1}+𝑢_{𝑖−1,𝑗−1}+𝑢_{𝑖+1,𝑗−1}+𝑢_{𝑖−1,𝑗+1}−4𝑢_𝑖𝑗2ℎ²−ℎ²21𝜕⁴𝑢𝜕𝑥²𝜕𝑦²+112𝜕⁴𝑢𝜕𝑥⁴+𝜕⁴𝑢𝜕𝑦⁴−𝑂ℎ⁴=𝑓_𝑖𝑗.(1.3) Based on the latter approximation, improved point and group iterative schemes have been developed over the last few years in solving several types of partial differential equations [1–5]. In particular, the Modified Explicit Decoupled Group (MEDG) method [6, 7] was formulated as the latest addition to this family of four-point explicit group methods in solving the Poisson equation. This method has been shown to be the fastest among the existing explicit group methods due to its lesser computational complexity.

Since it is well known that preconditioners play a vital role in accelerating the convergence rates of iterative methods, several preconditioned strategies have been used for improving the convergence rate of the explicit group methods derived from the standard and skewed (rotated) finite difference operators [8–11]. In particular, Saeed and Ali [12–14] presented an (𝐼+𝑆)-type preconditioning matrix applied to the systems obtained from the four-point Explicit Decoupled Group (EDG) and the Modified Explicit Decoupled Group (MEDG) methods for solving the elliptic partial differential equation, where 𝑆 is obtained by taking the first upper diagonal groups of the iteration matrix of the original system. The numerical experiments performed on these methods were seen to yield very encouraging results. However, no detailed studies of the spectral radius analysis of all these preconditioned systems have been done to confirm the superiority of this preconditioner.

The focus of this study is to establish the convergence properties of the preconditioned systems based on the splitting-type preconditioner (𝐼+𝑆) for improving the performance and reliability of this family of explicit group methods derived from the rotated finite-difference formula. We will prove that this type of preconditioner applied to the MEDG SOR can minimize the most the spectral radius of the preconditioned matrix provided the relaxation parameter is in a certain optimum range. This paper is organised as follows: in Section 2, we give a presentation of the preconditioner applied to the system resulted from the EDG SOR method. A brief description of the application of the preconditioner in block formulation to the MEDG SOR is given in Section 3. The theoretical convergence analysis of these methods is discussed in Section 4. In Section 5, we give a numerical example to confirm the results obtained in Section 4. Finally, we report a brief conclusion in Section 6.

2. Preconditioned Explicit Decoupled Group SOR (EDG SOR)

For convenience, we will now briefly explain some of the definitions used in this paper.

Definition 2.1 (see [15]). A matrix 𝐴 of order 𝑛 has property 𝐴 if there exists two disjoint subsets 𝑆 and 𝑇 of 𝑊={1,2,…,𝑛} such that if 𝑖≠𝑗 and if either 𝑎_𝑖𝑗≠0 and 𝑎_𝑖𝑗≠0, then 𝑖∈𝑆 and 𝑗∈𝑇 or else 𝑖∈𝑇 and 𝑗∈𝑆.

Definition 2.2 (see [3]). An ordered grouping 𝜋 of 𝑊={1,2,…,𝑛} is a subdivision of 𝑊 into disjoint subsets 𝑅₁,𝑅₂,…,𝑅_𝑞 such that 𝑅₁+𝑅₂+⋯+𝑅_𝑞=𝑊. Given a matrix 𝐴 and an ordered grouping 𝜋, we define the submatrices 𝐴_𝑚,𝑛 for 𝑚,𝑛=1,2,…𝑞 as follows: 𝐴_𝑚,𝑛 is formed from 𝐴 deleting all rows except those corresponding to 𝑅_𝑚 and all columns except those corresponding to 𝑅_𝑛.

Definition 2.3 (see [3]). Let 𝜋 be an ordered grouping with 𝑞 groups. A matrix 𝐴 has Property 𝐴^(𝜋) if the 𝑞×𝑞 matrix 𝑍=(𝑧_𝑟,𝑠) defined by 𝑧_𝑟,𝑠={0 if 𝐴_𝑟,𝑠=0 or 1 if 𝐴_𝑟,𝑠≠0} has Property 𝐴.

Definition 2.4 (see [15]). A matrix 𝐴 of order 𝑛 is consistently ordered if for some t there exist disjoint subsets 𝑆₁,𝑆₂,…,𝑆_𝑡 of 𝑊={1,2,…,𝑛} such that ∑^𝑡𝑘=1𝑆_𝑘=𝑊 and such that if 𝑖 and 𝑗 are associated, then 𝑗∈𝑆_𝑘+1 if 𝑗>𝑖 and 𝑗∈𝑆_𝑘−1 if 𝑗<𝑖, where 𝑆_𝑘 is the subset containing i.
Note that a matrix 𝐴 is a 𝜋-consistently ordered matrix if the matrix 𝑍 in Definition 2.3 is consistently ordered.
From the discretisation of the EDG finite-difference formula in solving the Poisson equation, the linear system 𝐴𝑢_∼=𝑏_∼(2.1) is obtained with [1] 𝐴=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣𝑅⁰𝑅₁𝑅₂𝑅₀𝑅₁𝑅₂⋱𝑅₀⋱⋱𝑅₁𝑅₂𝑅₀⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦_{(𝑁−1)2/2×(𝑁−1)2/2},𝑅₀=⎡⎢⎢⎢⎢⎢⎢⎣𝑅⁰⁰𝑅₀₁𝑅₀₂𝑅⋱₀₀⋱⋱𝑅₀₁𝑅₀₂𝑅₀₀⎤⎥⎥⎥⎥⎥⎥⎦_{(𝑁−1)×(𝑁−1)}𝑅₀₀,=⎡⎢⎢⎣−11−4141⎤⎥⎥⎦,𝑅⁰¹=⎡⎢⎢⎣−01040⎤⎥⎥⎦,𝑅⁰²𝑅₁=𝑅^𝑇01,=⎡⎢⎢⎢⎢⎢⎢⎣𝑅⁰¹𝑅₀₁𝑅₀₁⋱⋱𝑅₀₁𝑅₀₁⎤⎥⎥⎥⎥⎥⎥⎦_{(𝑁−1)×(𝑁−1)},𝑅₂=⎡⎢⎢⎢⎢⎢⎢⎣𝑅⁰²𝑅₀₂𝑅₀₂⋱𝑅⋱₀₂𝑅₀₂⎤⎥⎥⎥⎥⎥⎥⎦_{(𝑁−1)×(𝑁−1)}.(2.2) Let 𝐴∈⊄^{𝑛𝑖,𝑛𝑖}_𝜋,𝑝, (𝑝=𝑁−1) be written as 𝐴=𝐷−𝐸−𝐹, where 𝐷=diag(𝐴₁₁,𝐴₂₂,…,𝐴_𝑝,𝑝), 𝐸 and 𝐹 are strict block lower triangular, and strict block upper triangular parts of 𝐴. Here, the diagonal entries 𝐴_𝑖𝑖 are nonsingular and ⊄^{𝑛𝑖,𝑛𝑖}_𝜋,𝑝 denotes the set of all matrices in ⊄^{𝑛𝑖,𝑛𝑖}_𝜋,𝑝 which is of the form (2.1) relative to some given block partitioning 𝜋. The block Jacobi iteration matrix is 𝐵_𝐽=𝐷⁻¹(𝐸+𝐹)=𝐿+𝑈, where 𝐿=𝐷⁻¹𝐸,𝑈=𝐷⁻¹𝐹, the block Gauss-Seidel iteration matrix is 𝐵_GS=(𝐼−𝐿)⁻¹𝑈, and the Block Successive Over-Relaxation method (BSOR) iteration matrix is 𝐵_ℓ𝑤=(𝐼−𝑤𝐿)⁻¹{(1−𝑤)𝐼+𝑤𝑈}.(2.3) Since the matrix 𝐴 of (2.1) is a 𝜋-consistently ordered and possesses property 𝐴^(𝜋), therefore, the theory of block SOR is valid for this iterative method [1].
The theoretical optimum relaxation factor 𝜔_𝑝 for implementing the group SOR iterative scheme can be computed from the formula: 𝜔_𝑝=21+√1−𝜌²(𝐽),(2.4) where 𝜌(𝐽) is the spectral radius of the group Jacobian iterative matrix. Yousif and Evans [16] gave a good estimate of the spectral radius for the EDG method: 𝜌(𝐽)=1−76𝜋²ℎ².(2.5) In an effort to further accelerate the convergence rates of this method, Saeed and Ali [12] applied a preconditioner 𝑃 to the linear system (2.1) and transformed it into an equivalent system: 𝑃𝐴𝑢_∼=𝑃𝑏_∼(2.6) with 𝑃=(𝐼+𝑆), where 𝐼 is the identity matrix which has the same dimension as 𝐴 while 𝑆 is obtained by taking the first upper diagonal groups of 𝑅₀ in the original system above as the following: 𝑆=⎡⎢⎢⎢⎢⎢⎢⎣𝑍¹𝑍₁⋱𝑍₁⎤⎥⎥⎥⎥⎥⎥⎦_{(𝑁−1)2/2×(𝑁−1)2/2},𝑍₁=⎡⎢⎢⎢⎢⎢⎢⎢⎣0^∼−𝑅₀₁0_∼⋱⋱−𝑅₀₁0_∼⎤⎥⎥⎥⎥⎥⎥⎥⎦_{(𝑁−1)×(𝑁−1)}.(2.7) Here, 0_∼ is a (2×2) null matrix.
The system (2.1) becomes 𝐼+𝑆𝐴𝑢_∼=𝐼+𝑆𝑏_∼.(2.8) Hence, we have the linear system of equations: 𝐴𝑢_∼=𝑏_∼(2.9) with 𝐴=𝐼+𝑆𝐴=𝐼−𝐿−𝑆𝐿−𝑈−𝑆+𝑆𝑈,_∼𝑏=𝐼+𝑆𝑏_∼.(2.10) The SOR iteration matrix of this scheme is called the Modified Block Successive Over-Relaxation iteration matrix (MBSOR) and is given by 𝐵_ℓ𝑤=𝐼−𝑤𝐿+𝑆𝐿⁻¹(1−𝑤)𝐼+𝑤𝑈−𝑆+𝑆𝑈.(2.11) The matrix 𝐴 of (2.9) is 𝜋-consistently ordered and possesses property 𝐴^(𝜋) [13].

3. Preconditioned-Modified Explicit Decoupled Group SOR (MEDG SOR)

Using the MEDG approximation formula in discretising the Poisson equation, the following system is obtained [6]: 𝐴_𝑚𝑢=𝑏_𝑚,(3.1) where 𝐴_𝑚=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣𝑅^𝑚0𝑅_𝑚1𝑅_𝑚2𝑅_𝑚0𝑅_𝑚1𝑅_𝑚2𝑅⋱_𝑚0⋱⋱𝑅_𝑚1𝑅_𝑚2𝑅_𝑚0⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦_{(𝑁−2)2/2×(𝑁−2)2/2}𝑅_𝑚0,=⎡⎢⎢⎢⎢⎢⎢⎣𝑅^𝑚00𝑅_𝑚01𝑅_𝑚02𝑅⋱_𝑚00⋱⋱𝑅_𝑚01𝑅_𝑚02𝑅_𝑚00⎤⎥⎥⎥⎥⎥⎥⎦_{(𝑁−2)×(𝑁−2)}𝑅_𝑚00,=⎡⎢⎢⎣−11−4141⎤⎥⎥⎦,𝑅^𝑚01=⎡⎢⎢⎣−01040⎤⎥⎥⎦,𝑅^𝑚02𝑅_𝑚1=𝑅^𝑇𝑚₀₁,=⎡⎢⎢⎢⎢⎢⎢⎣𝑅^𝑚01𝑅_𝑚01𝑅_𝑚01⋱⋱𝑅_𝑚01𝑅_𝑚01⎤⎥⎥⎥⎥⎥⎥⎦_{(𝑁−2)×(𝑁−2)}𝑅_𝑚2,=⎡⎢⎢⎢⎢⎢⎢⎣𝑅^𝑚02𝑅_𝑚02𝑅_𝑚02⋱𝑅⋱_𝑚02𝑅_𝑚02⎤⎥⎥⎥⎥⎥⎥⎦_{(𝑁−2)×(𝑁−2)}.(3.2) It is observed that the partitioning of 𝐴_𝑚 is in the following block form: 𝐴_𝑚=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣𝐴^𝑚11𝐴_𝑚12𝐴_𝑚21𝐴_𝑚22𝐴_𝑚23𝐴_𝑚32𝐴⋱_𝑚33⋱⋱𝐴_{𝑚(𝑝−1)𝑝}𝐴_{𝑚𝑝(𝑝−1)}𝐴_𝑚𝑝𝑝⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦(3.3) with 𝑝=(𝑁−2), where 𝐴_𝑚𝑖𝑖∈⊄^{𝑛𝑖,𝑛𝑖}_𝜋,𝑝, 𝑖=1,2,…,𝑝, and ∑^𝑝𝑖=1𝑛_𝑖=𝑛. Let 𝐴_𝑚=𝐷_𝑚−𝐸_𝑚−𝐹_𝑚, where 𝐷_𝑚=diag(𝐴_𝑚11,𝐴_𝑚22,...,𝐴_𝑚𝑝𝑝) and 𝐸_𝑚=𝐸_𝑚𝑖𝑗=−𝐴^𝑚𝑖𝑗0ffoorr𝑗𝑗≥<𝑖𝑖𝐹_𝑚=𝐹_𝑚𝑖𝑗=−𝐴^𝑚𝑖𝑗0ffoorr𝑗𝑗≤>𝑖𝑖(3.4) are block matrices consisting of the block diagonal, strict block lower triangular, and strict block upper triangular parts of 𝐴_𝑚. Here, the diagonal entries 𝐴_𝑚𝑖𝑖 are nonsingular. The block Jacobi iteration matrix is 𝐵_𝐽(𝐴_𝑚)=𝐷^𝑚−1(𝐸_𝑚+𝐹_𝑚)=𝐿_𝑚+𝑈_𝑚, where 𝐿_𝑚=𝐷^𝑚−1𝐸_𝑚, 𝑈_𝑚=𝐷^𝑚−1𝐹_𝑚, while the block Gauss-Seidel iteration matrix is 𝐵_GS(𝐴_𝑚)=(𝐼_𝑚−𝐿_𝑚)⁻¹𝑈_𝑚. The Block Successive Over-Relaxation method (BSOR) iteration matrix is, therefore, 𝑇_ℓ𝑤=𝐼_𝑚−𝑤𝐿_𝑚⁻¹(1−𝑤)𝐼_𝑚+𝑤𝑈_𝑚.(3.5) Since the matrix 𝐴_𝑚 of (3.3) is 𝜋-consistently ordered and possesses property 𝐴^(𝜋), the theory of block SOR is also valid for this iterative method and, therefore, is convergent [6].

Similarly, the theoretical optimum relaxation factor 𝜔_𝑝 for implementing this group SOR-iterative scheme can be obtained from (2.4). In view of the fact that the grid spacing ℎ_MEDG=2ℎ_EDG, an estimate of the spectral radius of the group Jacobian iterative matrix of the MEDG method may be obtained from (2.5) as 𝜌(𝐽)=1−143𝜋²ℎ².(3.6) Good agreement between the theoretical estimates and experimental values of the optimum relaxation parameters was observed in our numerical experiments. Upon applying the left-sided preconditioner 𝑃=(𝐼+𝑆) to system (3.1), the following system is obtained [13]: 𝑃𝐴_𝑚𝑢=𝑃𝑏_𝑚(3.7) with 𝑆=⎡⎢⎢⎢⎢⎢⎢⎣𝑠¹𝑠₁⋱𝑠₁⎤⎥⎥⎥⎥⎥⎥⎦_{(𝑁−2)2/2×(𝑁−2)2/2},𝑠₁=⎡⎢⎢⎢⎢⎢⎢⎢⎣0^∼−𝑅_𝑚010_∼⋱⋱−𝑅_𝑚010_∼⎤⎥⎥⎥⎥⎥⎥⎥⎦_{(𝑁−2)×(𝑁−2)},(3.8) where 0_∼ is a (2×2) null matrix.

The preconditioner 𝐼+𝑆 is of the following form: 𝐼+𝑆=⎡⎢⎢⎢⎢⎢⎢⎣𝑠²𝑠₂⋱𝑠₂⎤⎥⎥⎥⎥⎥⎥⎦_{(𝑁−2)2/2×(𝑁−2)2/2},𝑠₂=⎡⎢⎢⎢⎢⎢⎢⎣𝐼⁰−𝑅_𝑚01𝐼₀⋱⋱−𝑅_𝑚01𝐼₀⎤⎥⎥⎥⎥⎥⎥⎦_{(𝑁−2)×(𝑁−2)}(3.9) Here, 𝐼₀ is a (2×2) identity matrix and the system (3.7) becomes 𝐼+𝑆𝐴^𝑚𝑢=𝐼+𝑆𝑏_𝑚.(3.10) Hence, 𝐴_𝑚𝑢=𝑏̃_𝑚,(3.11) where 𝐴_𝑚=𝐼+𝑆𝐴_𝑚=𝐼_𝑚−𝐿_𝑚−𝑆𝐿_𝑚−𝑈_𝑚−𝑆+𝑆𝑈_𝑚𝑏_𝑚̃,=𝐼+𝑆𝑏_𝑚.(3.12) The SOR iteration matrix will result in an Improved Modified Block Successive Over-Relaxation iteration matrix (IMBSOR) and is given by 𝑇_ℓ𝑤=𝐼_𝑚−𝑤𝐿_𝑚+𝑆𝐿_𝑚⁻¹(1−𝑤)𝐼^𝑚+𝑤𝑈_𝑚−𝑆+𝑆𝑈_𝑚.(3.13)

4. Convergence Properties of the Preconditioned Group Methods

In this section, we will derive several properties related to the convergence of the preconditioned methods discussed in Sections 2 and 3. We will begin with the presentation of several preliminary relevant theorems and lemmas which are needed for the proof of the convergence properties. The spectral radius of a matrix is denoted by 𝜌(⋅), which is defined as the largest of the moduli of the eigenvalues of the iteration matrix.

Theorem 4.1 (see [15]). If 𝐴=𝑀−𝑁 is a regular splitting of the matrix 𝐴 and 𝐴⁻¹≥0, then 𝜌𝑀⁻¹𝑁=𝜌𝐴⁻¹𝑁1+𝜌𝐴⁻¹𝑁<1.(4.1) Thus, an iterative method with coefficient matrix 𝑀⁻¹𝑁 is convergent for any initial vector 𝑥⁽⁰⁾.

An accurate analysis of convergence properties of the SOR method is possible if the matrix 𝐴 is consistently ordered in the following sense (see [17]).

Definition 4.2. A matrix 𝐴 is a generalized (𝑞,𝑟)-consistently ordered matrix (a GCO(𝑞,𝑟)-matrix) if Δ=det(𝛼^𝑞𝐸+𝛼^−𝑟𝐹−𝑘𝐷) is independent of 𝛼 for all 𝛼≠0 and for all 𝑘. Here, 𝐷=diag𝐴 and 𝐸 and 𝐹 are strictly lower and strictly upper triangular matrices, respectively, such that 𝐴=𝐷−𝐸−𝐹.

Definition 4.3 (see [17]). A matrix 𝐴 of the form (3.3) is said to be generally consistently ordered (𝜋,𝑞,𝑟) or simply GCO(𝜋,𝑞,𝑟), where 𝑞 and 𝑟 are positive integers, if for the partitioning 𝜋 of 𝐴, the diagonal submatrices 𝐴_𝑖𝑖,𝑖=1,2,…,𝑝(≥2), are nonsingular, and the eigenvalues of 𝐵_𝐽(𝛼)=𝛼^𝑟𝐿+𝛼^−𝑞𝑈(4.2) are independent of 𝛼, for all 𝛼≠0, where 𝐿 and 𝑈 are strict block lower and upper triangular parts of 𝐴 respectively.
For any matrix 𝐶=(𝑐_𝑖𝑗) in ⊄^{𝑛𝑖,𝑛𝑖}_𝜋,𝑝, let |𝐶| denote the block matrix in ⊄^{𝑛𝑖,𝑛𝑖}_𝜋,𝑝 with entries|𝑐_𝑖,𝑗|. Given the matrix 𝐵_𝐽=𝐿+𝑈,(4.3) then 𝜇 denotes the spectral radius of the matrix: ||𝐵_𝐽||=||𝐿+𝑈||,(4.4) so that 𝜇=𝜌||𝐵_𝐽||.(4.5)

Lemma 4.4 (see [17]). Let |𝐵_𝐽| of (4.4) be a GCO (𝑞,𝑟)-matrix and 𝑝=𝑞+𝑟. Then, for any real nonnegative constant 𝛼, 𝛽, and 𝛾 with 𝛾≠0 satisfying: 𝛼^𝑟𝛽^𝑞𝜇^𝑝<𝛾^𝑝, the matrix 𝐴^=𝛾𝐼−𝛼|𝐿|−𝛽|𝑈| is such that 𝐴^−1≥0.

Lemma 4.5 (see [14]). Suppose 𝐴=𝐼−𝐿−𝑈 is a GCO(𝜋,𝑞,𝑟), where −𝐿 and −𝑈 are strictly lower and upper triangular matrices, respectively. Let 𝐵_ℓ𝑤 be the block iteration matrix of the SOR method given by (2.3). If 0<𝑤<2, then the block SOR method converges, that is, 𝜌(𝐵_ℓ𝑤)<1.

Theorem 4.6 (see [14]). Suppose 𝐴=𝐼−𝐿−𝑈 is a GCO(𝜋,𝑞,𝑟), where −𝐿 and −𝑈 are strictly lower and upper triangular matrices, respectively. Let 𝐵_ℓ𝑤 and 𝐵_ℓ𝑤 be the iteration matrices of the SOR method given by (2.3) and (2.11), respectively. If 0<𝑤<2, then (i)𝜌(𝐵_ℓ𝑤)<𝜌(𝐵_ℓ𝑤) if 𝜌(𝐵_ℓ𝑤)<1,(ii)𝜌(𝐵_ℓ𝑤)=𝜌(𝐵_ℓ𝑤) if 𝜌(𝐵_ℓ𝑤)=1,(iii)𝜌(𝐵_ℓ𝑤)>𝜌(𝐵_ℓ𝑤) if 𝜌(𝐵_ℓ𝑤)>1.Using the results and definitions stated above, we can prove the following lemma and theorems in relation to the spectral radius of the iteration matrices of the preconditioned group methods and their unpreconditioned counterparts.

Lemma 4.7. Suppose 𝐴_𝑚=𝐼_𝑚−𝐿_𝑚−𝑈_𝑚 is a GCO(𝜋,𝑞,𝑟), where −𝐿_𝑚 and −𝑈_𝑚 are strictly lower and upper triangular matrices, respectively. Let 𝑇_ℓ𝑤 be the block iteration matrix of the SOR method given by (3.5). If 1≤𝑤<2, then the block SOR method converges, that is, 𝜌(𝑇_ℓ𝑤)<1.

Proof. Let the matrix 𝐴_𝑚 with partitioning 𝜋 be given as in (3.3) and let the block SOR iteration matrix 𝑇_ℓ𝑤 be given as in (3.5).
Set 𝐵^ℓ_𝑤=𝐼−||𝑤𝐿_𝑚||⁻¹||1−𝑤||𝐼_𝑚+|𝑤|||𝑈_𝑚||.(4.6) Clearly, we can see that |𝑇_ℓ𝑤|<𝐵^ℓ_𝑤 and hence we can conclude that 𝜌(𝑇_ℓ𝑤)≤𝜌(𝐵^ℓ_𝑤).
Consider the matrix 𝐴′∈⊄^{𝑛𝑖,𝑛𝑖}_𝜋,𝑝 defined by 𝐴′=𝑀_𝑚−𝑁_𝑚,(4.7) where 𝑀_𝑚=𝐼_𝑚−|𝑤||𝐿_𝑚| and 𝑁_𝑚=|1−𝑤|𝐼_𝑚+|𝑤||𝑈_𝑚|. It is easily seen that 𝑀_𝑚 is nonsingular and 𝐵^ℓ_𝑤=𝑀^1𝑚𝑁_𝑚. Moreover, since 𝑀^1𝑚≥0 and 𝑁_𝑚≥0, 𝑀_𝑚−𝑁_𝑚 is a regular splitting of 𝐴′ (cf.[11]). For 𝑤 satisfying the condition 1≤𝑤<2, Lemma 4.4 implies that 𝐴′⁻¹≥0. Therefore, recalling Theorem 4.1 above, we have 𝜌(𝐵^ℓ_𝑤)<1. Hence, 𝜌(𝑇_ℓ𝑤)<1, which completes the proof.

The result of Lemma 4.7 enables us to prove the following theorem

Theorem 4.8. Suppose 𝐴_𝑚=𝐼_𝑚−𝐿_𝑚−𝑈_𝑚 is a GCO(𝜋,𝑞,𝑟), where −𝐿_𝑚 and −𝑈_𝑚 are strictly lower and upper triangular matrices, respectively. Let 𝑇_ℓ𝑤 and 𝑇_ℓ𝑤 be the iteration matrices of the SOR method given by (3.5) and (3.13), respectively. If 1≤𝑤<2, then (i)𝜌(𝑇_ℓ𝑤)<𝜌(𝑇_ℓ𝑤) if 𝜌(𝑇_ℓ𝑤)<1,(ii)𝜌(𝑇_ℓ𝑤)=𝜌(𝑇_ℓ𝑤) if 𝜌(𝑇_ℓ𝑤)=1,(iii)𝜌(𝑇_ℓ𝑤)>𝜌(𝑇_ℓ𝑤) if 𝜌(𝑇_ℓ𝑤)>1.

Proof. From Lemma 4.7 and since the matrix 𝐴_𝑚 of (3.3) is a GCO(𝜋,𝑞,𝑟) and 𝑇_ℓ𝑤=(𝐼−𝑤𝐿_𝑚)⁻¹{(1−𝑤)𝐼_𝑚+𝑤𝑈_𝑚}, there exists a positive vector 𝑦 such that 𝑇_ℓ𝑤𝑦=𝜆𝑦,(4.8) where  𝜆=𝜌(𝑇_ℓ𝑤) or equivalently (1−𝑤)𝐼_𝑚+𝑤𝑈_𝑚𝑦=𝜆𝐼_𝑚−𝑤𝐿_𝑚𝑦.(4.9) Also, since 𝑇_ℓ𝑤=𝐼_𝑚−𝑤𝐿_𝑚+𝑆𝐿_𝑚⁻¹(1−𝑤)𝐼^𝑚+𝑤𝑈_𝑚−𝑆+𝑆𝑈_𝑚,(4.10) we can write 𝑇_ℓ𝑤𝑦−𝜆𝑦=𝐼_𝑚−𝑤𝐿_𝑚+𝑆𝐿_𝑚−𝜆𝐼_𝑚⁻¹(1−𝑤)𝐼^𝑚+𝑤𝑈_𝑚−𝑆+𝑆𝑈_𝑚−𝑤𝐿_𝑚+𝑆𝐿_𝑚𝑦.(4.11) Rearrange (4.11), we can get 𝑇_ℓ𝑤𝑦−𝜆𝑦=𝐼_𝑚−𝑤𝐿_𝑚+𝑆𝐿_𝑚−𝜆𝐼_𝑚⁻¹(1−𝑤)𝐼^𝑚+𝑤𝑈_𝑚+𝜆𝐿_𝑚−𝑤𝑆𝐿_𝑚+𝑤𝑆𝑈^𝑚+𝐼_𝑚𝑦.(4.12) But from (4.9), we have 𝜆𝑤𝐿_𝑚+𝑤𝑈_𝑚𝑦=(𝜆−1+𝑤)𝐼_𝑚𝑦.(4.13) Therefore, (4.12) can be written as 𝑇_ℓ𝑤𝑦−𝜆𝑦=𝐼_𝑚−𝑤𝐿_𝑚+𝑆𝐿_𝑚⁻¹𝑤𝑆𝐼^𝑚+𝜆𝐿_𝑚+𝑈_𝑚𝑦.(4.14) Hence, for 1≤𝑤<2 and from [10], we can get(i)𝜆<1, then 𝑇_ℓ𝑤𝑦−𝜆𝑦<0 and from Theorem 4.6 we have 𝜌(𝑇_ℓ𝑤)<𝜌(𝑇_ℓ𝑤),(ii)𝜆=1, then 𝑇_ℓ𝑤𝑦=𝜆𝑦 and from Theorem 4.6 we have 𝜌(𝑇_ℓ𝑤)=𝜌(𝑇_ℓ𝑤)=1,(iii)𝜆>1, then 𝑇_ℓ𝑤𝑦−𝜆𝑦>0 and from Theorem 4.6 we have 𝜌(𝑇_ℓ𝑤)>𝜌(𝑇_ℓ𝑤).
Thus, the proof is complete.

Theorem 4.9. Suppose 𝐴=𝐼−𝐿−𝑈 and 𝐴_𝑚=𝐼_𝑚−𝐿_𝑚−𝑈_𝑚 are GCO(𝜋,𝑞,𝑟), where −𝐿, −𝐿_𝑚, −𝑈 and −𝑈_𝑚 are strictly lower and upper triangular matrices of 𝐴 and 𝐴_𝑚, respectively. Let 𝐵_ℓ𝑤, 𝐵_ℓ𝑤, 𝑇_ℓ𝑤 and 𝑇_ℓ𝑤 be the iteration matrices of the SOR method given by (2.3), (2.11), (3.5), and (3.13), respectively. If 1≤𝑤<2, then (i)𝜌(𝑇_ℓ𝑤)<𝜌(𝑇_ℓ𝑤)<𝜌(𝐵_ℓ𝑤)<𝜌(𝐵_ℓ𝑤) if 𝜌(𝐵_ℓ𝑤)<1,(ii)𝜌(𝑇_ℓ𝑤)=𝜌(𝑇_ℓ𝑤)=𝜌(𝐵_ℓ𝑤)=𝜌(𝐵_ℓ𝑤) if 𝜌(𝐵_ℓ𝑤)=1,(iii)𝜌(𝑇_ℓ𝑤)>𝜌(𝑇_ℓ𝑤)>𝜌(𝐵_ℓ𝑤)>𝜌(𝐵_ℓ𝑤) if 𝜌(𝐵_ℓ𝑤)>1.

Proof. In the same manner of the proof of Theorem 4.8 and since the matrix 𝐴 of (2.9) is a GCO(𝜋,𝑞,𝑟), see [13], and 𝐵_ℓ𝑤={𝐼−𝑤(𝐿+𝑆𝐿)}⁻¹[(1−𝑤)𝐼+𝑤(𝑈−𝑆+𝑆𝑈)], there exists a positive vector 𝑣 such that 𝐵_ℓ𝑤𝑣=𝜆𝑣,(4.15) where 𝜆=𝜌𝐵_ℓ𝑤.(4.16) Equation (4.15) can be written as (1−𝑤)𝐼+𝑤𝑈−𝑆+𝑆𝑈𝑣=𝜆𝐼−𝑤𝐿+𝑆𝐿𝑣.(4.17) Also, since 𝑇_ℓ𝑤=(𝐼_𝑚−𝑤𝐿_𝑚)⁻¹{(1−𝑤)𝐼_𝑚+𝑤𝑈_𝑚}, we can write 𝑇_ℓ𝑤𝑣−𝜆𝑣==𝐼𝐼_𝑚𝑚−𝑤𝐿_𝑚⁻¹(1−𝑤)𝐼^𝑚+𝑤𝑈_𝑚−𝜆𝐼_𝑚−𝑤𝐿_𝑚𝑣−𝑤𝐿_𝑚⁻¹1−𝑤−𝜆𝐼_𝑚+𝑤𝜆𝐿_𝑚+𝑈_𝑚𝑣.(4.18) But, from (4.17) we have 𝑤𝜆𝐿+𝑈+𝑆𝜆𝐿+𝑈−𝐼𝑣=𝜆−1+𝑤𝐼𝑣.(4.19) Thus, from (4.19) and since 𝐴_𝑚 of (3.3) is a GCO(𝜋,𝑞,𝑟) matrix, we can get 𝑤𝜆𝐿_𝑚+𝑈_𝑚+𝑆𝜆𝐿_𝑚+𝑈_𝑚−𝐼_𝑚𝑣=𝜆−1+𝑤𝐼_𝑚𝑣.(4.20) Equation (4.18) can then be written as𝑇_ℓ𝑊𝑣−𝜆𝑣=𝐼_𝑚−𝑤𝐿_𝑚⁻¹𝑤𝑆−𝜆𝐿_𝑚−𝑈_𝑚−𝐼_𝑚𝑣.(4.21) Hence, we can conclude that, for 1≤𝑤<2, if(a)𝜆<1, then 𝑇_ℓ𝑤𝑣−𝜆𝑣<0 and from Lemma 4.7 we have 𝜌(𝑇_ℓ𝑤)<𝜌(𝐵_ℓ𝑤),(b)𝜆=1, then 𝑇_ℓ𝑤𝑣=𝜆𝑣 and from Lemma 4.7 we have 𝜌(𝑇_ℓ𝑤)=𝜌(𝐵_ℓ𝑤)=1,(c)𝜆>1, then 𝑇_ℓ𝑤𝑣−𝜆𝑣>0 and from Lemma 4.7 we have 𝜌(𝑇_ℓ𝑤)>𝜌(𝐵_ℓ𝑤).
In consequence of the above, for 1≤𝑤<2 and from Theorems 4.6 and 4.8, we have(i)𝜌(𝑇_ℓ𝑤)<𝜌(𝑇_ℓ𝑤)<𝜌(𝐵_ℓ𝑤)<𝜌(𝐵_ℓ𝑤) if 𝜌(𝐵_ℓ𝑤)<1,(ii)𝜌(𝑇_ℓ𝑤)=𝜌(𝑇_ℓ𝑤)=𝜌(𝐵_ℓ𝑤)=𝜌(𝐵_ℓ𝑤) if 𝜌(𝐵_ℓ𝑤)=1,(iii)𝜌(𝑇_ℓ𝑤)>𝜌(𝑇_ℓ𝑤)>𝜌(𝐵_ℓ𝑤)>𝜌(𝐵_ℓ𝑤) if 𝜌(𝐵_ℓ𝑤)>1,and the theorem is proved.

In view of Theorem 4.9, the superiority of the preconditioned MEDG SOR over the unpreconditioned MEDG SOR, EDG SOR methods and also preconditioned EDG SOR are confirmed for certain relaxation parameters lying in an optimum range.

5. Numerical Experiments and Discussion of Results

To further confirm the results obtained in Theorems 4.8 and 4.9, several experiments were carried out on the following model problem with Dirichlet boundary conditions: 𝜕²𝑢𝜕𝑥²+𝜕²𝑢𝜕𝑦²𝑢(𝑥,0)=1,𝑢(0,𝑦)=1,=𝑥𝑢²(𝑥+,1𝑦)²=𝑒𝑒^𝑥𝑦_𝑥,,𝑢(1,𝑦)=𝑒^𝑦.(5.1) This problem has an exact solution 𝑢(𝑥,𝑦)=𝑒^𝑥𝑦 with the unit square as the solution domain. The values of 𝑢 were calculated using different mesh sizes, 34, 86, 118, 186, and 222. The tolerance was set to be 𝜀=5×10⁻⁶. The experimental optimum relaxation parameter 𝑤 was obtained by running the programs repeatedly and choosing the values which gave the fastest rate of convergence. The computer processing unit was Intel(R) Core(TM) 2Duo with memory of 3Gb and the software used to implement and generate the results was Developer C++ Version 4.9.9.2. Tables 1 and 2 display the corresponding number of iterations (𝑘), optimum execution times (𝑡), and the maximum errors (𝑒) for the unpreconditioned and preconditioned methods of EDG SOR and MEDG SOR, respectively.

Table 1 

Comparison of performances for the original EDG SOR and MEDG SOR.

Table 2 

Comparison of performances for the Preconditioned EDG SOR and MEDG SOR.

From the results in Table 1, it is obvious that the original MEDG SOR method is superior to the EDG SOR method in terms of the number of iterations and computing times. The superiority of the preconditioned MEDG SOR over the preconditioned EDG SOR was also depicted in Table 2. The preconditioned EDG SOR was also outperformed by the unpreconditioned MEDG as shown in Figure 1 since the spectral radius of the latter is smaller than the former as proven in Theorem 4.9. From the numerical results, it is also apparent that the preconditioned MEDG SOR scheme requires the least computing effort amongst the four methods in terms of number of iterations and execution times due to its smallest spectral radius value amongst the four schemes.

Figure 1 

Number of iterations (k) for the four methods for different mesh sizes N.

Figure 1 shows the number of iterations needed for convergence for the unpreconditioned and preconditioned methods which were shown to be in agreement with the theoretical results obtained in Theorem 4.9.

6. Conclusion

In this paper, we present a theoretical convergence analysis of a specific splitting-type preconditioner in block formulation applied to the linear systems resulted from a class of group iterative schemes specifically the EDG SOR and the MEDG SOR schemes. We have shown that the spectral radius of the iteration matrix of the preconditioned MEDG SOR method is the smallest compared to the unpreconditioned MEDG SOR, EDG SOR, and preconditioned EDG SOR methods provided that the relaxation parameter 𝜔∈[1,2). This work confirms the superiority of the preconditioned MEDG SOR method theoretically and experimentally in terms of convergence rates among this class of group iterative methods.

Acknowledgment

The authors acknowledge the Fundamental Research Grant Scheme (203/PMATHS/6711188) for the completion of this article.