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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 487273, 12 pages
Convergence Analysis of an Iterative Method for Nonlinear Partial Differential Equations
Department of Mathematics, National Kaohsiung Normal University, Yanchao District, Kaohsiung City 82444, Taiwan
Received 22 May 2013; Accepted 20 July 2013
Academic Editor: E. Karapinar
Copyright © 2013 Hung-Yu Ke et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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 this method joining traditional monotone iterative method (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.
In terms of solving linear equations, we usually use two different iterative methods, namely, the Jacobi and Gauss-Seidel methods [1–3]. The monotone iterative (MI) schemes which combine linear iterative techniques, respectively, are presented and analyzed in [4–8] 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 [9–12]. It is also useful for numerical solutions of these types of problems approximated, for instance, by the finite difference [5, 6, 13–15], finite element , 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 . 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 . 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 is of fundamental importance . For parabolic problems with time delays we refer to . In addition, we also utilize MI schemes to handle nonlinear problems on analysis of numerical results for semiconductor equations [21–23] and the Poisson Boltzmann equation .
Consider the nonlinear boundary-value problem: in a two-dimensional domain with boundary , where is the outward normal derivative of on , , , are positive functions on , and are nonnegative functions on with , and and are given functions in their respective domains. For the nonlinear function , we give two assumptions:(i)(ii)if , then there exists a function , such that for all we have
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 , where , , and are the diagonal, lower-off-diagonal, 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 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 , the SORMI method generates a sequence of iterates , , 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 . 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.
2. A Finite Difference Discretization
We discuss problem (1) in a rectangular domain . Let , , and let , for , . The set of points in and are defined, respectively, by and . When no confusion arises we write a point in by . Define The finite difference method for differential and boundary operators in (1) leads to a discrete system in the form for all , where the coefficients , , , , and are associated with the diffusion coefficients , , as well as the boundary coefficients and , is associated with the boundary functions , and Typical choice of the coefficients in (5) for the interior mesh points is given by (e.g., see ). For the boundary points , where or , the above coefficients are associated with the boundary coefficients and , and possess the property (for the case ), Furthermore, strict inequality in (8) holds for at least one when the boundary condition is not of pure Neumann condition. In either case, these coefficients satisfy the condition Condition (9) is our basic hypothesis for the boundedness and the convergence of the SORMI sequence. Now we rewrite system (5) in a compact form:
Definition 1. A real matrix with for all and for all is an -matrix if is nonsingular, and .
Obviously, is a diagonally dominant with strict inequality for at least one . Since the domain is connected, the above property implies that is nonsingular, and . Hence, is an -matrix. This implies that for any nonnegative diagonal matrix , exists and is nonnegative.
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 for all , . If is the null matrix and , we say that is a nonnegative (positive) matrix. Since column vectors are 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 if and only if is nonsingular and is nonnegative, where is the spectral radius of .
Remark 3. In reality, the four conditions in the definition of -matrix are somewhat redundant, and equivalent conditions that are more rigorous will be (i) for all , (ii) is nonsingular, and (iii) . The condition, for all , is implied by the other three. Moreover, let be the diagonal of , and . We can also obtain . A comparison theorem is as follows.
Let and be two -matrices, with . Then we have .
Remark 4. Let us look in more detail at the algebraic system (10) [26, 27]. The connectedness assumption of ensures that is irreducible. Condition (9) implies that is irreducibly diagonally dominant . Let , where is the diagonal matrix of . It can be shown that , using Perron-Frobenius theorem and the theory of regular splittings. A theorem states the following.
If is a real matrix with for all , then the following are equivalent.(i) is nonsingular, and .(ii)The diagonal entries of are positive real numbers. is nonnegative, irreducible, and convergent.Thus, we know that is a diagonally dominant -matrix.
3. The SORMI Method
We now arrive to construct the SORMI sequence.
Definition 5. A vector with components is called an upper solution of (10) if and is called a lower solution of (10) if We say that and are ordered if . Given any ordered upper and lower solutions , , we set Define where is any nonnegative scalar satisfying , and and are the components of and , respectively. Then problem (10) is equivalent to Suppose that can be written in the splitting form , where , , and are the diagonal, lower-off-diagonal, and upper-off-diagonal matrices of , respectively. The elements of are positive, and those of and are nonnegative. Given an initial iterate vector , the SOR method for solving the linear system is where is a relaxation factor, and . Moreover, the Gauss-Seidel MI method for solving the nonlinear system (15) is defined by Thus, we define the SORMI method for solving the nonlinear system (15) by
4. 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 solutions , of (10), let , be two vectors with components, and . Then
Proof . Let and be the components of and , respectively. By the mean value theorem, where lies between and . From (14), we have . Hence This completes the proof.
In [2, page 83], the theorem is stated as follows.
Theorem 7. If is an matrix, then if and only if is nonsingular, and , where is a spectral radius of .
Hence, we quote the above theorem to obtain the following lemma.
Lemma 8. The matrix in (18) is nonsingular, and is nonnegative for .
Definition 9. Let the vector . We define , .
Definition 10. Let . The vector 2-norm is usually defined by .
Definition 11. Let be a vector 2-norm. The induced matrix 2-norm of an matrix is defined by Let be a sequence generated by the SORMI method with initial vector , where , are upper and lower solutions of (10), respectively. We have the following.
Lemma 12. for any . In fact, .
Proof. Let , , . Since , for all , . Hence, that is, for any .
To prove that the SORMI sequences are bounded, we must define several values about matrix and vector norms.
Notations and Assumptions. (a) Let . By Lemma 12, (b) In , we assume that is uniformly bounded for , and is known boundary value; that is, there exists such that for , where and are mesh sizes. By Definition 10, (5) and (10), we have (c) Define and assume that (d) Consider two vectors: Since , , , and , we know that is a nonnegative vector. It follows that . Let Then (e) Define
Proof. By (28) and (30), we can choose a constant .
Consider two cases of the sequence .
Case 1. Let . Then By (27) and , we obtain Similarly, and thus So . By Lemma 12 and (28), we obtain Case 2. Let . Then By (26), we have Consider the iterative process. An induction argument gives Hence, from Definition 11, , (24), (25), (26), (30), and , we obtain Thus, by Cases 1 and 2, there exists , such that for all , and the proof is completed.
5. The Convergence of the SORMI Sequences
In , we assume that if , then there is a function , such that . From Theorem 13, we have shown the boundedness of the SORMI sequences; that is, there exists a constant such that . Let satisfy . So for all . Let . exists and . Hence, we can find the constant . So we have Then we obtain the inequality:
Theorem 14. Let , be a pair of ordered upper and lower solutions of (10), respectively, and let be a sequence generated by the SORMI method with initial vector . Suppose that where and are defined by (26). Then the sequence is convergent for .
Proof. By Theorem 13, is bounded for . Hence, , such that for all . Since , and , then, for every , , such that for all . Let . By (37), we have Hence, from and (41), we obtain Inductively we have Hence, We have proved that is a Cauchy sequence for . It implies that is convergent for .
Furthermore, we provide another proof about the convergence of the SORMI sequences for without the assumptions and . Denote the sequence by when and by when , and refer to them as the maximal and minimal sequences, respectively. The following theorem gives some monotone property of these sequences.
Theorem 15. The maximal and minimal sequences and given by (18) with and possess the monotone property Moreover for each , and are ordered upper and lower solutions.
Proof. We will use induction to complete the proof of monotone property. First, let . From (11), (18), and ,
By Lemma 8, we obtain . This leads to . Similarly let , and use (12) to obtain
Since , it implies that . Secondly, let . By (18),
We have from Lemma 6, , and the nonnegative property of that . It follows from Lemma 8 again that . The above conclusions imply that
We finally assume that for some . Let , and by (18) we have
Since , we have . So which shows that . Similarly, let and simultaneously. We see that
By , we have , and . Hence, , and . The proof of monotone property (46) is completed.
To show that is an upper solution for each , we observe from (18) that By and (46), we obtain By Lemma 6, we have This shows that is an upper solution for each . Similarly, we have that is, for each , is a lower solution, and thus the proof is completed.
Theorem 16. Let , be a pair of ordered upper and lower solutions of (10). Then the sequences , given by (18) with and converge monotonically to solutions and of (10), respectively, where Moreover, and if is any solution in , then .
Proof. By Theorem 15, the limits and as exist, and the relation (58) also holds. Letting in (18) shows that and are solutions of (15). The equivalence between (10) and (15) ensures that and are solutions of (10). Now if is a solution in , then and are ordered upper and lower solutions. Using and , Theorem 15 implies that for every . Letting gives . A similar argument using and as ordered upper and lower solutions yields . This proves the theorem.
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 . However, if , where is the smallest positive eigenvalue of and then the following theorem holds.
Proof. Let and and be the components of and , respectively. By (58), we have . On the other hand, by the mean value theorem, we have
where lies between and . Hence,
Case 1. If . From the form of , is strictly diagonally dominant. It follows that , and thus .
Case 2. If . Since is the smallest positive eigenvalue of , then is the biggest positive eigenvalue of . From Theorem 7 and , we have Hence, . So we know that . 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 (see Theorems 15, 16, and 17). However, the nonnegative property of is not available when . 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 , , and 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.
6. 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 . Now we present some numerical results with two test problems.
Example 19. Consider the one-dimensional boundary value problem: The exact solution is Let , and choose , . Then, Hence we choose . We examine the assumptions and 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 . 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.
Example 20. As the second test problem, consider the logistic model in population growth problem with nonlinear function
where is a positive constant, and is a possible internal source . The discretized function is given by
For physical reasons, we suppose that , , and , such that is a lower solution of (10). For upper solution, consider the following.
Case 1. and . (i) If the upper solution is dependent on , define , , and . Then is an upper solution, where and are any upper bounds of and , respectively. (ii) If upper solution is independent of , then is an upper solution, where , and is the vector with components which satisfies .
Case 2. (i) If , then is an upper solution, where is a constant which satisfies . (ii) If , then is an upper solution.
We now give a model problem where the exact solution is known explicitly . This problem is given by It is easy to verify that when and the exact solution of (71) is given by Since for , the constant pair and are ordered upper and lower solutions whenever with . Hence, (71) is case 2 whenever , and we have the upper and lower solutions and . Let . We define If , , then