Advances in Numerical Analysis

Volume 2016, Article ID 2161279, 11 pages

http://dx.doi.org/10.1155/2016/2161279

## Preconditioning and Uniform Convergence for Convection-Diffusion Problems Discretized on Shishkin-Type Meshes

^{1}School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland^{2}Department of Mathematical Sciences, Kent State University at Stark, 6000 Frank Avenue NW, North Canton, OH 44720, USA

Received 5 September 2015; Revised 9 December 2015; Accepted 26 January 2016

Academic Editor: Yinnian He

Copyright © 2016 Thái Anh Nhan and Relja Vulanović. 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.

#### Abstract

A one-dimensional linear convection-diffusion problem with a perturbation parameter multiplying the highest derivative is considered. The problem is solved numerically by using the standard upwind scheme on special layer-adapted meshes. It is proved that the numerical solution is -uniform accurate in the maximum norm. This is done by a new proof technique in which the discrete system is preconditioned in order to enable the use of the principle where “-uniform stability plus -uniform consistency implies -uniform convergence.” Without preconditioning, this principle cannot be applied to convection-diffusion problems because the consistency error is not uniform in . At the same time, the condition number of the discrete system becomes independent of due to the same preconditioner; otherwise, the condition number of the discrete system before preconditioning increases when tends to 0. We obtained such results in an earlier paper, but only for the standard Shishkin mesh. In a nontrivial generalization, we show here that the same proof techniques can be applied to the whole class of Shishkin-type meshes.

#### 1. Introduction

In this paper we consider a convection-diffusion boundary-value problem in which the coefficient of the diffusion term is a small positive parameter . This makes the problem a singularly perturbed one. Singularly perturbed boundary-value problems are characterized by the presence of the perturbation parameter which multiplies the highest derivative in the differential equation considered. Therefore, when tends to 0, the degree of the differential equation is reduced and the solution of the reduced equation typically does not satisfy all the boundary conditions imposed. This is why the solution of the original problem may have boundary and/or interior layers; that is, the solution may change abruptly over narrow regions whose size depends directly on . It is typical that the derivatives of the solution behave in the layers like , where is an appropriate positive number. Because of this, the classical numerical methods for solving boundary-value problems do not work well for singularly perturbed problems and special numerical methods need to be constructed [1–4]. Ideally, the goal of these methods is to achieve -uniform pointwise convergence. The layer-adapted meshes of Shishkin or Bakhvalov types are frequently used to meet this goal.

The interest in singular perturbation problems is motivated by their numerous applications. For instance, the introduction in [4] mentions Navier-Stokes equations with large Reynolds number and other convection-diffusion equations modeling water-pollution problems, oil-extraction processes, flows in chemical reactors, convective heat-transport problems with large Péclet numbers, and semiconductor devices. Let us also mention the application to a steady, fully developed laminar flow of alumina-water nanofluid [5].

In this paper, we consider the simplest singularly perturbed convection-diffusion problem in one dimension:where and , , and are -functions. The problem has a unique solution in [6, 7]. In general, this solution exhibits an exponential boundary layer near .

Problem (1) serves as a model problem for discussing some theoretical aspects related to the proof of -uniform pointwise convergence of numerical methods for singularly perturbed convection-diffusion problems. This proof is not as simple as for the singularly perturbed reaction-diffusion problems, where the proof can be based on the following classical principle, which originates from nonperturbed problems (cf. [8–10]).

*Principle 1. *-uniform stability and -uniform consistency, both in the maximum norm, imply -uniform pointwise convergence.

Principle 1 does not work in the proof of -uniform pointwise convergence for convection-diffusion problem (1). When (1) is discretized by using the upwind finite-difference scheme on the Shishkin mesh, a careful analysis shows that the consistency error in the layer part is not -uniformly accurate in the maximum norm (cf. [11, Sec. 6.1.3]). Hence, special techniques like barrier functions [1–4] or the use of hybrid stability inequalities [2, 9, 12, 13] are required to prove the -uniform pointwise convergence.

We show in [14] that it is possible to apply Principle 1 to problems of type (1) after the linear system, which results when (1) is discretized, is preconditioned. The upwind discretization scheme on the standard Shishkin mesh is considered in that paper. Although the scheme is -uniformly stable, the discrete linear system is ill-posed since its condition number grows unboundedly when , as pointed out by Roos in [15]. By using a relatively simple diagonal preconditioner, Roos eliminates the -dependence of the condition number.

It is shown in [14] that a suitable modification of Roos’ preconditioner enables the proof, based on Principle 1, that the pointwise errors of the numerical solution are -uniformly of almost first order. Since the standard Shishkin mesh is the only one considered in [14], the question is now how to use the same approach and achieve the same result for the general class of Shishkin-type meshes in the sense of [2, 16]. The present paper addresses this question.

Therefore, for convection-diffusion problem (1), we consider the new proof technique of -uniform pointwise convergence, introduced in [14], and we generalize it to the whole class of Shishkin-type meshes. The generalization is not entirely straightforward since the general properties of the meshes require many new technical details. Moreover, we slightly modify the preconditioner used in [14].

In this final paragraph of the introduction, we give the outline of the paper. In Section 2, we describe the Shishkin-type meshes which are used when discretizing problem (1). We analyze the conditioning of the upwind finite-difference discretization in Section 3. Then, in Section 4, we define the preconditioner and give a proof of -uniform stability of the preconditioned discrete system. We arrive at the main result, the proof of -uniform pointwise convergence using Principle 1, in Section 5. The last section, Section 6, contains some final conclusions.

#### 2. Shishkin-Type Meshes

Let be a general discretization mesh with points , whereWe consider the Shishkin-type (or S-type) meshes as introduced in [16]; see [2] as well. Any such mesh is dense in the boundary layer near and transitions to a coarse uniform mesh at a point constructed like in the standard Shishkin mesh:Here, is a user-chosen positive parameter and we think of as of , although a modification of this value can also be used, like in [17]. We assume that ; otherwise, the values of are unreasonably large.

The coarse part of the mesh is obtained by uniformly dividing the interval into subintervals of length , where is a positive integer such that satisfies and , and where, from this point on, denotes a generic positive constant independent of both the perturbation parameter and the mesh parameter . Inside the layer, the mesh points are formed by a* mesh-generating* function , which is a monotonically increasing function satisfying and . In other words, S-type meshes are defined bywhere .

The mesh steps are defined as ; the steps for belong to the fine part of the mesh and for . Let also , and let mesh functions on be denoted by , , and so forth. For a function defined on , indicates and stands for the corresponding mesh function. We identify mesh functions with -dimensional column vectors, . We use the maximum norm of :The matrix norm induced by the above maximum vector norm is also denoted by .

Following [2, 16], let us define the* mesh-characterizing* function which is related to the mesh-generating function as follows:This function is monotonically decreasing on with and .

We need to assume some further properties of the mesh.

*Assumption 1. *Let the mesh-generating function be piecewise differentiable such thatMoreover, we also assume that fulfillswhere .

*Remark 2. *Examples of mesh-generating functions satisfying the above assumptions are the standard Shishkin mesh (S-mesh), Bakhvalov-Shishkin mesh (BS-mesh), and Vulanović-Shishkin mesh (VS-mesh). They are summarized in Table 1.