Fitted Numerical Methods For Singular Perturbation Problems

The following sections are included:

• Preface

• Notation, Terminology and Acknowledgments

• Contents

One of the main reasons for studying singularly perturbed differential equations is because, in many instances, the partial differential equations of hydrodynamics are singularly perturbed. Indeed, it is this feature of the equations that explains, theoretically, the physical phenomenon of boundary layers. The presence of boundary layers in fluid and gas dynamics was not known before the early years of the twentieth century, when Prandtl gave his seminal paper [Prandtl (1904)] to the Third International Congress of Mathematicians, which revolutionised the theoretical understanding of many flow phenomena…

The following sections are included:

• Linear reaction-diffusion equation

• Linear convection-diffusion equation

• Burger’s equation

In this chapter the concept of an ɛ-uniform method is introduced and discussed in general terms. This serves to introduce the main ideas, which, in subsequent chapters, are applied to problems of increasing generality…

In this chapter ɛ-uniform fitted operator methods are constructed for a modified form of each of the first three problems described in Chapter 2. The modification, in each case, is the replacement of the homogeneous differential equation by an inhomogeneous one having a non-zero function on its right hand side. The convergence properties of fitted operator methods for these modified problems can then be discussed in a meaningful way. Note that such a discussion would be trivial for the three original problems with constant coefficients and homogeneous equations, because the fitted operator methods described below are, by design, exact for these simple problems. The fitted operator methods consist of fitted finite difference operators on uniform meshes. To construct these methods the following uniform meshes and finite difference operators are needed…

To introduce the idea of fitted mesh methods the problems discussed in Chapter 2 are considered again here. In all cases a piecewise uniform fitted mesh turns out to be sufficient for the construction of an ɛ-uniform method. Of course, more complicated meshes may also be used, but the simplicity of piecewise uniform meshes is considered to be one of their major attractions. Furthermore, piecewise uniform meshes turn out to be adequate for handling a surprisingly wide variety of singularly perturbed problems…

The ɛ-uniform convergence of the solutions of a fitted mesh method for the linear reaction-diffusion problem in one dimension is established in this chapter. An essentially second order ɛ-uniform error estimate is obtained. The problem considered is the following second order self adjoint problem with a variable coefficient.

Discrete two point boundary value problems, using upwind finite difference operators on fitted piecewise uniform meshes of the following kind, are now considered

In this chapter the ɛ-uniform convergence of the solutions of a fitted mesh method, for a linear convection-diffusion problem in one dimension, is established. The problem considered is the following second order non selfadjoint problem with a variable coefficient…

The numerical methods discussed in this chapter differ from those in other chapters in several important respects. First, they are finite element methods with standard finite element subspaces, which are shown to be ɛ- uniform provided that the mesh is fitted appropriately. This demonstrates that there are ɛ-uniform fitted mesh methods based on centred difference operators, and on other operators for which there is no maximum principle, at least in the one dimensional case. Secondly, because there is no discrete maximum principle, the ɛ-uniform error estimates can not be established as easily as in the previous chapter. Furthermore, the theoretical results described here have not been extended to higher dimensions, unlike the situation for the finite difference methods discussed in Chapters 6 and 8. The method of proof of these theoretical results is based on the techniques presented in [Stynes and O’Riordan (1991)]. More general results for such problems may be found in [Sun and Stynes (1995)]. For an up-to-date account of finite element methods for singular perturbation problems the reader is referred to the books [Roos et al. (2008)] and [Morton (1995)]…

In higher dimensions a wide range of singularities may appear in the solutions of singular perturbation problems. These singularities can be caused by, for example, a lack of smoothness in the data, irregularities in the geometry of the domain or the presence of mixed derivatives in the differential equations. For such problems the Schwarz iterative method is useful, because it permits the decomposition of the solution domain into a finite number of overlapping subdomains, in each of which a different numerical method may be used. In each subdomain the numerical method may be chosen so that it is appropriate for the singularities arising there. A further benefit of such a decomposition is that it opens the door to the use of parallel processing algorithms. A particularly nice feature of the combination of the Schwarz iterative method with the piecewise uniform fitted mesh methods described in this book, is that in each subdomain the numerical method has a uniform mesh. Here, when the Schwarz iterative method is applied to a problem concerning a differential equation, it is referred to as a continuous Schwarz method, while it is called a discrete Schwarz method if the problem concerns a finite difference equation…

Two dimensional problems are considered for the first time in this chapter. These are much closer to practical applications than one dimensional problems, which were considered above mainly for illustrative purpose. Even in the linear case the variety of singular perturbation phenomena, that can arise in the solutions of linear convection-diffusion problems in two dimensions, is surprisingly rich. Linearisations of the Navier-Stokes equations frequently give rise to linear convection-diffusion problems…

In this chapter bounds are established on the derivatives of the solution of a linear convection-diffusion problem in two dimensions, in the case where only regular boundary layers are present. These bounds are required in the derivation of the ɛ-uniform error estimates in the next chapter…

In this chapter the two-dimensional linear convection-diffusion problem (P_ɛ) from Chapter 12 is considered again. The ɛuniform convergence of a fitted mesh method is established for the case when only regular boundary layers occur. The numerical method, used to solve (P_ɛ); is the standard upwind finite difference operator on the piecewise uniform mesh where is the piecewise uniform fitted mesh on the edge E_1,2, condensing at the boundary point x₁ = 1, and is the piecewise uniform fitted mesh on the edge E_4,1, condensing at the boundary point x₂ = 1. The transition parameters are chosen to satisfy, for i = 1 and i = 2,

The purpose of this chapter is to prove that, for a simple problem involving a parabolic boundary layer, it is not possible to construct an ɛ-uniform finite difference method based on a uniform rectangular mesh and a fitted finite difference operator of a particular form. The restriction of the finite difference operator to a particular form is made in the interests of simplicity, but the proof given here can be extended to much more general cases…

In this chapter it is shown that, for a simple problem involving both initial and parabolic boundary layers, it is not possible to construct an ɛ-uniform finite difference method based on a rectangular fitted mesh and a standard upwind finite difference operator.

The following sections are included:

• Appendix A
  • Some a priori Bounds for Differential Equations in Two Dimensions

• Bibliography

• Index