Numerical Solution of Quasilinear Singularly Perturbed Problems by the Principle of Equidistribution

In this paper, the numerical solution and its error analysis of quasilinear singular perturbation two-point boundary value problems based on the principle of equidistribution are given. On the non-uniform grid of the uniformly distributed arc-length monitor function, the solution of the simple upwind scheme is obtained. It is proved that the adaptive simple upwind scheme based on the principle of equidistribution has uniform convergence for small perturbation parameters. Numerical experiments are carried out and the error analysis are confirmed.


Introduction
We consider a quasilinear singularly perturbed two-point boundary value problem: x b x u C ≤ ≤ and Journal of Applied Mathematics and Physics solution u, and u has an exponential boundary layer at 0 x = (see [1] [2] [3]).
Linß considered the problem (1) with in [4]. The problem was discretized using a simple upwind finite difference scheme on adapted meshes using grid equidistribution of monitor functions. Kopteva and Stynes considered the same case in [5]. In [6], Jugal and Srinivasan considered a linear problem and proved the uniform first-order convergence of the numerical solution on adapted meshes by equidistributing the arc-length monitor function. In [7], Linß, Roos and Vulanović considered the problem (1), which was discretized using a nonstandard upwinded first-order difference scheme on generalized Shishkin-type mesh. Linß considered the problem (1) on Bakhvalov-type mesh in [8], on BS-type mesh in [9]. Zheng, Li, and Gao proposed a class of hybrid difference schemes with variable weights on Bakhvalov-Shishkin mesh in [10] and the parameter-uniform second-order convergence of approximating to the solution and the derivative on BS-type mesh and that of nearly second-order on Shishkin mesh were proved.
In this paper, we consider the problem (1) in the general case. The problem is discretized using a simple upwind finite difference scheme on adapted meshes using grid equidistribution of the arc-length monitor function. We will prove the uniform first-order convergence and the numerical experiments verify our analysis.

Simple Upwind Scheme and a Posteriori Error Estimate
We set: So, the problem (1) is equivalent to: For any nodes on interval [0, 1], The simple upwind scheme is: For any grid function ( ) v x , we define the norm: Easily get:

Journal of Applied Mathematics and Physics
Define the linear case of the problem (1): where v and w be two arbitrary functions.
To simplify some arguments, we assume that: The operators L and T are related as follow: Lemma 1: Suppose that operator L satisfies inequality (7), then equation (5) has a unique solution Lemma 3: There exists a constant C such that From Lemma 3, we get: The approach to prove the above lemmas is similar to that in [2], where

First-Order Uniform Convergence
The adaptive mesh method based on the principle of equal arc-length distribution is applied here.
Define the discrete operator N L of operator L by ( ) For any mesh function we have the representation Thus, from (10) (11), we have ( ) According to Lemma 3 and (9) (10), we have ( )

Numerical Experiment
Example 1 (see [11]): We study the following quasilinear singularly perturbed two-point boundary value problem: The exact solution of this problem is: For any grid function { } i v we define the following discrete maximum norm: In order to verify the error estimate, the numerical convergence rate of the solution is calculated as follows: 2 2 log , where, we use the maximum node error :  Table 1.
From the numerical example, we can see that the adaptive mesh can converge faster and be more accurate when the number of grid points is small in Figure 1,

Conclusion
In this paper, we consider the general problem (1). A simple upwind difference scheme is used to discretize on adaptive meshs based on grid equidistribution of the arc-length monitor function. The uniform first-order convergence of the algorithm is proved, and numerical experiments verify our analysis.