Robust Monotone Iterates for Nonlinear Singularly Perturbed Boundary Value Problems
© Igor Boglaev. 2009
Received: 8 April 2009
Accepted: 11 May 2009
Published: 14 May 2009
This paper is concerned with solving nonlinear singularly perturbed boundary value problems. Robust monotone iterates for solving nonlinear difference scheme are constructed. Uniform convergence of the monotone methods is investigated, and convergence rates are estimated. Numerical experiments complement the theoretical results.
We are interested in numerical solving of two nonlinear singularly perturbed problems of elliptic and parabolic types.
where is a positive parameter, and is sufficiently smooth function. For this problem is singularly perturbed, and the solution has boundary layers near and (see  for details).
where is a positive parameter. Under suitable continuity and compatibility conditions on the data, a unique solution of this problem exists. For problem (1.2) is singularly perturbed and has boundary layers near the lateral boundary of (see  for details).
In the study of numerical methods for nonlinear singularly perturbed problems, the two major points to be developed are: (i) constructing robust difference schemes (this means that unlike classical schemes, the error does not increase to infinity, but rather remains bounded, as the small parameter approaches zero); (ii) obtaining reliable and efficient computing algorithms for solving nonlinear discrete problems.
Our goal is to construct a -uniform numerical method for solving problem (1.1), that is, a numerical method which generates -uniformly convergent numerical approximations to the solution. We use a numerical method based on the classical difference scheme and the piecewise uniform mesh of Shishkin-type . For solving problem (1.2), we use the implicit difference scheme based on the piecewise uniform mesh in the -direction, which converges -uniformly .
A major point about the nonlinear difference schemes is to obtain reliable and efficient computational methods for computing the solution. The reliability of iterative techniques for solving nonlinear difference schemes can be essentially improved by using component-wise monotone globally convergent iterations. Such methods can be controlled every time. A fruitful method for the treatment of these nonlinear schemes is the method of upper and lower solutions and its associated monotone iterations . Since an initial iteration in the monotone iterative method is either an upper or lower solution, which can be constructed directly from the difference equation without any knowledge of the exact solution, this method simplifies the search for the initial iteration as is often required in the Newton method. In the context of solving systems of nonlinear equations, the monotone iterative method belongs to the class of methods based on convergence under partial ordering (see [5, Chapter 13] for details).
The purpose of this paper is to construct -uniformly convergent monotone iterative methods for solving -uniformly convergent nonlinear difference schemes.
The structure of the paper is as follows. In Section 2, we prove that the classical difference scheme on the piecewise uniform mesh converges -uniformly to the solution of problem (1.1). A robust monotone iterative method for solving the nonlinear difference scheme is constructed. In Section 3, we construct a robust monotone iterative method for solving problem (1.2). In the final Section 4, we present numerical experiments which complement the theoretical results.
2. The Elliptic Problem
The following lemma from  contains necessary estimates of the solution to (1.1).
where constant is independent of .
For , the boundary layers appear near and .
2.1. The Nonlinear Difference Scheme
where . We now formulate a discrete maximum principle for the difference operator and give an estimate of the solution to (2.4).
- (i)If a mesh function satisfies the conditions(2.5)
If , then the following estimate of the solution to (2.4) holds true:
where , .
The proof of the lemma can be found in .
2.2. Uniform Convergence on the Piecewise Uniform Mesh
In the following theorem, we give the convergence property of the difference scheme (2.3).
where constant is independent of and .
From Lemma 2.1, the following estimate on holds:
From (2.3), (2.15), by the mean-value theorem, we conclude that satisfies the difference problem
Using the assumption on from (1.1) and (2.24), by (2.6), we prove the theorem.
2.3. The Monotone Iterative Method
In this section, we construct an iterative method for solving the nonlinear difference scheme (2.3) which possesses monotone convergence.
where is the residual of the difference scheme (2.3) on .
where . In view of the maximum principle in Lemma 2.2, we conclude the required inequality.
The following theorem gives the monotone property of the iterative method (2.27).
and the sequences converge at the linear rate .
where , . Since the mesh function is nonpositive on and taking into account (2.26), we conclude that is an upper solution. By induction on , we obtain that , , , and prove that is a monotonically decreasing sequence of upper solutions.
We now prove that the monotone sequence converges to the solution of (2.3). Similar to (2.33), we obtain
By (2.6), which leads to the uniqueness of the solution to (2.3). This proves the theorem.
Since and is nonnegative, we conclude that is an upper solution.
Taking into account that as , where is the solution to (2.3), we conclude the theorem.
From Theorems 2.3 and 2.5 we conclude the following theorem.
where and constant is independent of and .
3. The Parabolic Problem
3.1. The Nonlinear Difference Scheme
We now formulate a discrete maximum principle for the difference operator and give an estimate of the solution to (3.3).
- (i)If a mesh function on a time level satisfies the conditions(3.4)
If , then the following estimate of the solution to (3.3) holds true:
where , .
The proof of the lemma can be found in .
3.2. The Monotone Iterative Method
where is the residual of the difference scheme (3.2) on .
This result can be proved in a similar way as for the elliptic problem.
The following theorem gives the monotone property of the iterative method (3.7).
We consider only the case of the upper sequence, and the case of the lower sequence can be proved in a similar way.
If is an upper solution, then from (3.7) we conclude that
and from (3.7), it follows that satisfies the boundary conditions.
Using the mean-value theorem and the equation for from (3.7), we represent in the form
where , . Since the mesh function is nonpositive on and taking into account (3.6), we conclude that is an upper solution to (3.2). By induction on , we obtain that , , , and prove that is a monotonically decreasing sequence of upper solutions.
We now prove that the monotone sequence converges to the solution of (3.2). The sequence is monotonically decreasing and bounded below by , where is any lower solution (3.9). Now by linearity of the operator and the continuity of , we have also from (3.7) that the mesh function defined by
By Lemma 3.1, which leads to the uniqueness of the solution to (3.2). This proves the theorem.
The functions , are upper and lower solutions, respectively. This result can be proved in a similar way as for the elliptic problem.
where is the solution to (3.2), , and constant is independent of , and .
where is defined in (3.18).
Introduce the notation
Using (3.6), (3.17), and the mean-value theorem, estimate from (3.7) by (3.5),
Similarly, from (3.2) and (3.19), it follows that
where all constants are independent of , and . Taking into account that , we prove the estimate (3.18) with .
where is the exact solution to (3.2), and constant is independent of , and . From here and Theorem 3.3, we conclude the following theorem.
where , and constant is independent of , and .
4. Numerical Experiments
It is found that in all numerical experiments the basic feature of monotone convergence of the upper and lower sequences is observed. In fact, the monotone property of the sequences holds at every mesh point in the domain. This is, of course, to be expected from the analytical consideration.
4.1. The Elliptic Problem
Our numerical experiments show that for and , iteration counts for monotone method (2.27) on the piecewise uniform mesh are independent of and , and equals 12 and 8 for the lower and upper sequences, respectively. These numerical results confirm our theoretical results stated in Theorem 2.5.
Numbers of iterations for the Newton iterative method.
4.2. The Parabolic Problem
For the parabolic problem (1.2), we consider the test problem with and . This problem gives , , and the initial lower and upper solutions are chosen in the form of (3.17).
Our numerical experiments show that for and , on each time level the number of iterations for monotone method (3.7) on the piecewise uniform mesh is independent of and and equal 4, 4, and 3 for , respectively. These numerical results confirm our theoretical results stated in Theorem 3.3.
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