Convergence Analysis of an Iterative Method for Nonlinear Partial Differential Equations

We will combine linear successive overrelaxation method with nonlinear monotone iterative scheme to obtain a new iterative method for solving nonlinear equations.The basic idea of thismethod joining traditionalmonotone iterativemethod (known as the method of lower and upper solutions) which depends essentially on the monotone parameter is that by introducing an acceleration parameter one can construct a sequence to accelerate the convergence. The resulting increase in the speed of convergence is very dramatic. Moreover, the sequence can accomplish monotonic convergence behavior in the iterative process when some suitable acceleration parameters are chosen. Under some suitable assumptions in aspect of the nonlinear function and the matrix norm generated from this method, we can prove the boundedness and convergence of the resulting sequences. Application of the iterative scheme is given to a logistic model problem in ecology, and numerical results for a test problem with known analytical solution are given to demonstrate the accuracy and efficiency of the present method.


Introduction
In terms of solving linear equations, we usually use two different iterative methods, namely, the Jacobi and Gauss-Seidel methods [1][2][3]. The monotone iterative (MI) schemes which combine linear iterative techniques, respectively, are presented and analyzed in [4][5][6][7][8] for solving nonlinear equations. The method of monotone iterations is a classical tool for the study of the existence of solutions of semilinear PDEs of certain types [9][10][11][12]. It is also useful for numerical solutions of these types of problems approximated, for instance, by the finite difference [5,6,[13][14][15], finite element [16], or boundary element [17,18] method. It is a constructive method that depends essentially on only one parameter, called the monotone parameter herein, which determines the convergent behavior of the iterative process. Besides, the block Picard, block Jacobi, and block Gauss-Seidel MI methods are also developed and compared the rates of convergence with the point MI schemes [6]. The block MI methods accelerate the rate of convergence more than the point MI methods. In particular, Ortega and Rheinboldt [19, page 456] mention an analysis of the Newton-SOR methods to research some properties of convergence for relaxation factor 0 < < 1. The MI methods have been widely used in the treatment of certain nonlinear parabolic and elliptic differential equations. For instance, in the study of certain subsonic flows and molecular interactions, the equation Δ = 2 is of fundamental importance [18]. For parabolic problems with time delays we refer to [20]. In addition, we also utilize MI schemes to handle nonlinear problems on analysis of numerical results for semiconductor equations [21][22][23] and the Poisson Boltzmann equation [24].
Consider the nonlinear boundary-value problem: − [(D (1) ) + (D (2) ) ] = ( , , ) , in Ω, Applying the finite difference method to (1), we obtain a system of nonlinear algebraic equations in a compact form: Suppose that A can be written in the splitting form = D−L−U, where D, −L, and −U are the diagonal, lower-offdiagonal, and upper-off-diagonal matrices of , respectively. We consider that linear SOR method can be combined with nonlinear MI scheme to obtain a nonlinear SOR monotone iterative method for solving nonlinear equations which gives rise to the terminology "SORMI". The basic idea of this method joining MI method which depends essentially on the monotone parameter Γ is that by introducing an acceleration parameter one can construct a sequence to accelerate the convergence. The algorithm is similar to the SOR method. Roughly speaking, given an initial vector (0) , the SORMI method generates a sequence of iterates { ( ) }, = 0, 1, . . ., by solving the equation: where is a relaxation factor. Under some suitable assumptions in aspect of the nonlinear function and the matrix norm generated from this method, we can prove the boundedness and convergence of the resulting sequences. Moreover, the sequences can accomplish monotonic convergence in the iterative process when some suitable relaxation factors are chosen.
The structure of the paper is as follows. In Section 2, we briefly make a description for discretization process to obtain algebraic equations for model (1) and state some properties of the matrix. Section 3 deals with the monotone parameter and constructs the SORMI scheme. We show the boundedness and convergence of the SORMI sequence in Sections 4 and 5. Moreover, we offer another proof for the convergence of the SORMI sequence in the case 0 < < 1. In Section 6, we solve a one dimensional problem, and a logistic model in population growth problem and numerical results of the method are also given to verify the theoretical analysis. The final section is for some concluding remarks.

Remark 2.
Nonnegative matrices play a crucial role in the theory of matrices. They are important in the study of convergence of iterative methods and arise in many applications including economics, queuing theory, and chemical engineering. Let = ( ) and = ( ) be two real × matrices. Then, If is the null matrix and ≥ (> ), we say that is a nonnegative (positive) matrix. Since column vectors are × 1 matrices, we will use the terms nonnegative and positive vector throughout. A theorem which has important consequences on the analysis of iterative methods should be stated. Let be a nonnegative matrix. Then ( ) < 1 if and only if − is nonsingular and ( − ) −1 is nonnegative, where ( ) is the spectral radius of .

Remark 3.
In reality, the four conditions in the definition of M-matrix are somewhat redundant, and equivalent conditions that are more rigorous will be (i) ≤ 0 for all ̸ = , (ii) is nonsingular, and (iii) −1 ≥ 0. The condition, > 0 for all , is implied by the other three. Moreover, let be the diagonal of , and ≡ − −1 . We can also obtain ( ) < 1. A comparison theorem is as follows.

Remark 4.
Let us look in more detail at the algebraic system (10) [26,27]. The connectedness assumption of Ω ensures that is irreducible. Condition (9) implies that is irreducibly diagonally dominant [28]. Let = D − B, where D is the diagonal matrix of . It can be shown that 0 < (B) < 1, using Perron-Frobenius theorem and the theory of regular splittings. A theorem states the following. If = ( ) is a real × matrix with ≤ 0 for all ̸ = , then the following are equivalent.
(ii) The diagonal entries of are positive real numbers.
B is nonnegative, irreducible, and convergent.
Thus, we know that is a diagonally dominant M-matrix.

The SORMI Method
We now arrive to construct the SORMI sequence.

The Boundedness of the SORMI Sequences
Before the convergence analysis of the method, we want to ask whether the SORMI sequences are bounded. Now we consider the property.
Lemma 6. Given a pair of upper and lower solutions̃,̂of (10), let U, V be two vectors with ( + 1) × ( + 1) components, and̃≥ ≥ ≥̂. Then Proof. Let and V be the components of and , respectively. By the mean value theorem, where lies between V and . From (14), we have ≥ + . Hence This completes the proof.
In [2, page 83], the theorem is stated as follows.
Hence, we quote the above theorem to obtain the following lemma.
To prove that the SORMI sequences are bounded, we must define several values about matrix and vector norms.
Proof. By (28) and (30), we can choose a constant Consider two cases of the sequence { ( ) }.
Journal of Applied Mathematics 7 Furthermore, we provide another proof about the convergence of the SORMI sequences for 0 < < 1 without the assumptions (H3) and (H4). Denote the sequence by when (0) =̂, and refer to them as the maximal and minimal sequences, respectively. The following theorem gives some monotone property of these sequences.
In Theorem 16, and are often called maximal and minimal solutions in ⟨̂,̃⟩, respectively. In general, these two solutions are not necessarily the same. Let be symmetric. Then has real and positive eigenvalues [2]. However, if < , where is the smallest positive eigenvalue of and where lies between and . Hence, Hence, ≤ 0. So we know that = 0. This proves = . The uniqueness follows from the relation ≥ * ≥ for any solution * ∈ ⟨̂,̃⟩.
Remark 18. For system (1), the well-known method of upper and lower solutions with SORMI is applied for the case 0 < < 1 (see Theorems 15,16,and 17). However, the nonnegative property of (1 − )(D + Γ) + U is not available when > 1. A new approach for solving this problem by the boundedness of the SORMI sequences and Cauchy sequence property is proposed. To make sure of the convergence of the SORMI sequences, the assumptions (H1), (H2), and (H3) are necessary. But it should be pointed out that these constraints are not easy to be verified. It is important to weaken these constraints when the SORMI method is applied to realistic problems. Fixed point theory is a powerful tool to overcome this problem for further study.

Numerical Results
Assume that the matrix of (10) is an × matrix. The componentwise SORMI algorithm is given as follows: Another equivalent form is The main requirement for the application of the various MI schemes is the existence of a pair of ordered upper and lower solutions. To ensure the existence, the nonlinear function must have some necessary conditions. Hence, in Section 1, we require that ( ) is uniformly bounded in R. Now we present some numerical results with two test problems.
Example 19. Consider the one-dimensional boundary value problem: The exact solution is Let ( , ) = (−1/2) + (5/2) sin( √ 2 ) − (1/2) sin( √ 2 ), and choose Γ = diag(2, 2, . . . , 2), (0) = (0, 0, . . . , 0) . Then, Hence we choose = 1/2. We examine the assumptions (H3) and (H4) and the convergence of the SORMI sequences. The numerical results are given in Table 1, and the exact and approximate solutions are shown in Figure 1. Moreover, we sketch the relation between the numbers of iterations and relaxation factors by Figure 2.     The numbers of iterations for the SORMI method are listed in Table 1. We focus on the SORMI method with = 1.9. The number of iterations is 127. Compared with it, the Jacobi and Gauss-Seidel MI methods require 4905 and 2645 iterations, respectively. The resulting increase in the speed of convergence is very dramatic. Moreover, the values of ‖ ‖ and in Table 1 are smaller than 1 which verify our theory of the boundedness and convergence for the SORMI method.
where is a positive constant, and ( , ) is a possible internal source [6]. The discretized function is given by For physical reasons, we suppose that ( , , ) ≥ 0, ( , ) ≥ 0, and ( , ) ≥ 0, such that̂= 0 is a lower solution of (10). For upper solution, consider the following. We now give a model problem where the exact solution is known explicitly [6]. This problem is given by It is easy to verify that when = 2 /4 and the exact solution of (71) is given by The SORMI method takes 94 iterations with = 1.92 when (0) = 0 is applied. The approximate solution is similar to the exact solution as shown in Figure 3. Since they are similar, we omit to sketch the exact solution figure. In comparison with the SORMI method, the Jacobi and Gauss-Seidel MI methods require 2243 and 1249 iterations, respectively. The reduction of iterations is significant. The numerical results are listed in   Table 2. The relation between the numbers of iterations and relaxation factors is shown in Figure 4. They verify our theory of the boundedness and convergence for the SORMI method again.

Conclusions
A new iterative method for solving nonlinear equations is developed in this paper. It combines SOR method with MI scheme and therefore gives rise to the terminology "SORMI. " The boundedness and convergence of the SORMI sequence are proven under some suitable assumptions. Some numerical examples are given to verify our theory of the boundedness and convergence for the SORMI method. Moreover, the reduction of iterations is quite significant in comparison with the Jacobi and Gauss-Seidel MI methods.