Quadratic convergence of monotone iterates for semilinear elliptic obstacle problems
 Jinping Zeng^{1},
 Haowen Chen^{2} and
 Hongru Xu^{3}Email author
https://doi.org/10.1186/s136600171513x
© The Author(s) 2017
Received: 13 March 2017
Accepted: 12 September 2017
Published: 25 September 2017
Abstract
In this paper, we consider the numerical solution for the discretization of semilinear elliptic complementarity problems. A monotone algorithm is established based on the upper and lower solutions of the problem. It is proved that iterates, generated by the algorithm, are a pair of upper and lower solution iterates and converge monotonically from above and below, respectively, to the solution of the problem. Moreover, we investigate the convergence rate for the monotone algorithm and prove quadratic convergence of the algorithm. The monotone and quadratic convergence results are also extended to the discrete problems of the twosided obstacle problems with a semilinear elliptic operator. We also present some simple numerical experiments.
Keywords
MSC
1 Introduction
Problem (1.1) has been widely applied in many scientific, engineering or economic problems, e.g., in the diffusion problems involving MichaelisMenten or second order irreversible reactions [1–5].
The numerical algorithms for solving problem (1.2) have been developing rapidly. Among these algorithms, there are a kind of monotone algorithms based on the upper or lower solutions of the problem. We refer to [7, 8, 10, 11] and the references therein for details. These algorithms can be regarded as extensions of the monotone algorithms for solving elliptic boundary value problems or their discretizations (see, e.g., in [6, 12–14]). In these algorithms, any generated iterate is an upper (or lower) solution sequence which then converges to the solution monotonically. In this paper, we extend the monotone iterative approach for elliptic boundary value equation, presented in [6], to complementarity as well as twosided obstacle problems. By using a pair of upper and lower solutions as two initial iterates, one can construct two monotone sequences which converge monotonically from above and below, respectively, to the solutions of the problems. Especially, the initial iterative solutions in the monotone iterative algorithms can be obtained directly by solving two discrete linear complementarity problems without any knowledge of the exact solution. Quadratic convergence is also proved for the algorithms.
The structure of the paper is as follows. In Section 2, we provide two procedures, in which only a pair of linear complementarity problems are needed to be solved. Following these procedures, we can obtain a pair of upper and lower solutions of the nonlinear complementarity problem we are concerned with. In Sections 3 and 4, we propose a monotone iterative algorithm and deal with the quadratic convergence of the monotone iterates, respectively. In Section 5, we extend the results obtained in Sections 24 to the twosided obstacle problem. In Section 6, we present some simple numerical experiments.
2 Upper and lower solutions and their initializations
The approach presented in this paper is based on the upper and lower solutions of the problems. In this section, we introduce the definitions of upper and lower solutions of problem (1.2) and discuss their properties.
Lemma 2.1
Proof
For the lower solution, we have a similar result as follows.
Lemma 2.2
Problems (2.2) and (2.5) can be regarded as lower and upper obstacle problems with variables ū and \(\underline{u}\), respectively. According to Lemmas 2.1 and 2.2, we can obtain a pair of upper and lower solutions of nonlinear complementarity problem (1.2) by solving two linear complementarity problems (2.2) and (2.5). As for linear complementarity problems, there are many classic and efficient iterative or direct algorithms to solve them. We refer to [11, 15–17] for further discussions.
3 Monotone iterative algorithm for complementarity problem
In this section, we propose an algorithm for solving the nonlinear complementarity problem (1.2) and discuss its monotone convergence. Firstly, we present the algorithm as follows.
Algorithm 3.1
Direct algorithms with a polynomial computational complexity are fruitful for subproblems (3.1). Especially, there are a lot of polynomial algorithms for linear complementarity problems with an Mmatrix. We refer to [15–19] for more details.
Remark 3.1
Lemma 3.1
Let \(u_{\alpha}^{(k)}\), \(\alpha=1,1\), be the solutions of problems (3.1), respectively. If \(u^{(k1)}_{1}\ge u^{(k1)}_{1}\), \(u_{1}^{(k)}\geq u_{1}^{(k)}\).
Proof
The following theorem gives the monotone convergence of Algorithm 3.1.
Theorem 3.1
Proof
4 Quadratic convergence rate of the monotone algorithm
Theorem 4.1
Proof
Theorem 4.2
Estimates (4.6) and (4.7) indicate that when \(f_{i}(v_{i})\) (\(i=1,2,\ldots, n\)) have convex (or concavity ) property in a neighborhood of the solution of problem (1.2), the maximal (or minimal) sequence, generated by (3.1), converges quadratically to the solution of the problem.
5 Extensions to a twosided obstacle problem
Obviously, ϕ and ψ are candidates of \(S_{1}\) and \(S_{1}\), respectively. In the following, we present two schemes, which can produce an upper or a lower solution of problem (5.1) from any \(w\in R^{n}\).
Scheme 5.1
 Step 1.:

Solve the following LCP of finding \(\bar{u}\ge\phi\) such thatwhere \(\Lambda^{*}\) and ḡ are the same as those given in Section 2.$$ \bigl(A+\Lambda^{*}\bigr)\bar{u}\ge\bar{g}, \qquad (\bar{u} \phi)^{T}\bigl[\bigl(A+\Lambda^{*}\bigr)\bar{u}\bar{g}\bigr]=0, $$(5.6)
 Step 2.:

Letand \(\Gamma=\operatorname{diag}(\tau_{1},\tau_{2},\ldots,\tau_{n})\). Define$$\tau_{i}= \textstyle\begin{cases} 0 & \mbox{if } \bar{u}_{i}\leq\psi_{i},\\ \psi_{i}\bar{u}_{i} & \mbox{if } \bar{u}_{i}> \psi_{i} \end{cases} $$where \(e\in R^{n}\) is the vector of ones.$$\tilde{\bar{u}}=\bar{u}+\Gamma e, $$
Thereby, we obtain the following result.
Lemma 5.1
Let \(\tilde{\bar{u}}\) be produced by Scheme 5.1. Then \(\tilde{\bar{u}}\in S_{1}\), where \(S_{1}\) is defined by (5.4).
Similarly, we can obtain a lower solution of the problem by the following scheme.
Scheme 5.2
 Step 1.:

Solve the following LCP of finding \(\underline{u}\leq\psi\) such thatwhere \(\Lambda^{*}\) and \(\underline{g}\) are the same as those given in Section 2.$$ \bigl(A+\Lambda^{*}\bigr)\underline{u}\leq\underline{g},\qquad (\underline{u}\psi )^{T}\bigl[\bigl(A+\Lambda^{*}\bigr)\underline{u}\bar{g}\bigr]=0, $$(5.7)
 Step 2.:

Letand \(\Gamma=\operatorname{diag}(\tau_{1},\tau_{2},\ldots,\tau_{n})\). Define$$\tau_{i}= \textstyle\begin{cases} 0 & \mbox{if } \bar{u}_{i}\geq\phi_{i},\\ \phi_{i}\underline{u}_{i} & \mbox{if } \bar{u}_{i}< \phi_{i} \end{cases} $$$$\tilde{\underline{u}}=\underline{u}+\Gamma e. $$
Similar to Lemma 5.1, we have the following result.
Lemma 5.2
Let \(\tilde{\underline{u}}\) be produced by Scheme 5.2. Then \(\tilde{\bar{u}}\in S_{1}\), where \(S_{1}\) is defined by (5.5).
By Schemes 5.1 and 5.2, we can obtain a pair of upper and lower solutions of twosided obstacle problem (5.1) by solving two affine upper and lower obstacle problems (5.6) and (5.7), instead of solving one twosided obstacle problem. To our knowledge, as for the twosided obstacle problem, direct algorithms with polynomial computational complexity are few.
Algorithm 5.1
Similar to Lemma 3.1, the following lemma holds.
Lemma 5.3
Let \(u_{\alpha}^{(k)}\), \(\alpha=1,1\), be the solutions of problems (5.1), respectively. If \(u^{(k1)}_{1}\ge u^{(k1)}_{1}\), \(u_{1}^{(k)}\geq u_{1}^{(k)}\).
According to Lemma 5.3, we have the following monotone and quadratic convergence similar to Theorems 3.1, 4.1 and 4.2.
Theorem 5.1
6 Numerical experiments
In this section, we present numerical experiments in order to investigate the performance of the proposed algorithms. The programs are coded in Visual C++ 6.0 and run on a computer with 2.0 GHz CPU. We consider the following two problems.
Problem 1
Problem 2
We compare different algorithms from the point of view of iteration numbers and cpu times (seconds). Here, we consider three algorithms: Algorithm 3.1, denoted by (AL); the semismooth equation approach proposed in [21], denoted by SSN; and primaldual algorithm proposed in [22], denoted by PDA.
Comparisons of ω
ω  iter  ω  iter  ω  iter 

0.9  225  1.1  152  1.3  101 
1.5  64  1.7  36  1.9  22 
Comparisons of iteration numbers and execution times for Problem 1
h  AL  SSN  PDA  

iter  cpu  iter  cpu  iter  cpu  
\(\frac{1}{10}\)  2  0.013  3  0.01  2  0.001 
\(\frac{1}{20}\)  2  0.678  3  0.603  2  0.012 
\(\frac{1}{40}\)  2  9.246  3  37.883  2  0.187 
\(\frac{1}{60}\)  2  237.486  3  434.057  2  0.932 
Comparisons of iteration numbers and execution times for Problem 2
h  AL  SSN  PDA  

iter  cpu  iter  cpu  iter  cpu  
\(\frac{1}{10}\)  2  0.1  5  0.019  1  0.001 
\(\frac{1}{20}\)  2  0.775  5  1.011  1  0.037 
\(\frac{1}{40}\)  2  12.768  7  89.629  4  8.653 
\(\frac{1}{60}\)  3  444.401  22  3,177.98  11  1,024.21 
From Tables 23, we can easily see that the iteration numbers of Algorithm 3.1 are stable, which may also mean that the initials obtained by (2.2) may be a good solution guess. While for SSN and PDA, the iteration numbers increase when the dimensions of Problem 2 become large. As we can see from Table 3, the algorithm we proposed seems to be more effective for solving largescale problems. The main reason may be as follows. Algorithm 3.1 takes only a few iterations for all problems, and in each iteration it only needs to solve two linear complementarity problems. These subproblems are solved by PSOR rapidly. For SSN, it takes a lot of time to solve the system of linear equations to obtain directions, especially for largescale problems. For PDA, by using the active set strategy, in each iteration, it only needs to solve a reduced system of linear equations, where the dimension is much less than the original one. Therefore, the execution time for PDA is also less compared with SSN.
7 Conclusions
In this paper, we have considered the numerical solution for the discretization of semilinear elliptic complementarity problems. Based on the upper and lower solutions of the problem, we have proposed a monotone algorithm and proved that iterates are a pair of upper and lower solution iterates and converge monotonically from above and below, respectively, to the solution of the problem. Moreover, we have established quadratic convergence of the algorithm. The limited numerical results showed that the proposed algorithm is effective.
Declarations
Acknowledgements
The work was supported by the NSF of China (Grant Nos. 11271069, 11601188) and by Training Program for Outstanding Young Teachers in Guangdong Province (Grant No. 20140202).
Authors’ contributions
All authors jointly worked on the results. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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