Fitted numerical methods for singular perturbation problems. Error estimates in the maximum norm for linear problems in one and two dimensions.

*(English)*Zbl 0915.65097
Singapore: World Scientific. xiv, 166 p. (1996).

From the authors’ preface: This is a book about numerical methods for solving singularly perturbed differential equations. It is a theoretical book but throughout the text the reader is referred to papers published in the literature which contain material about the implementation of these methods and the results of extensive computations on test problems. These numerical methods are simple to describe and easy to implement. It is the proof of the theoretical results that is difficult. The topic of the present book is the theory for linear problems in one dimension and in two dimensions when the solutions have only regular layers. Since most of the ideas and techniques presented in this monograph have not previously been published in detail in book form, its goal is to explain them in a reasonably simple way. Therefore no attempt is made to be comprehensive nor to state and prove the results in the most general case. Instead the key ideas are explained for simple problems containing the crucial difficulties.

This book falls naturally into three parts. The first three chapters provide motivation and an elementary introduction to some aspects of the subject. The next seven chapters are concerned with problems exclusively in one dimension, while in the final five chapters problems in two spatial dimensions or in one spatial dimension and time are considered.

In the first three chapters simple examples of various one-dimensional problems involving singular perturbations are described and some issues concerning their numerical solution are discussed. The fact that such problems cannot be solved numerically in a completely satisfactory manner by standard methods is then explained. This indicates the need for methods that behave uniformly well whatever the value of the singular perturbation parameter. Such methods are called \(\varepsilon\)-uniform methods, where \(\varepsilon\) is the singular perturbation parameter.

Simple examples of \(\varepsilon\)-uniform finite difference methods are presented in Chapters 4 and 5. These are of two kinds: the first are the fitted operator methods which comprise specially designed finite difference operators on standard meshes; the second are the fitted mesh methods which comprise standard finite difference operators on specially designed meshes.

In Chapter 4 fitted operator methods on uniform meshes are described for some simple problems in one dimension. The chapter concludes with the construction of the EL-Mistikawy-Werle fitted operator method for linear convection-diffusion equations in one dimension and with a modern proof that it is an \(\varepsilon\)-uniform method. In Chapter 5 fitted mesh methods for simple problems in one dimension are constructed. A simple basic lemma is established and it is then proved that fitted mesh methods for initial value problems are \(\varepsilon\)-uniform. The proof that fitted mesh methods for linear reaction-diffusion equations in one dimension are \(\varepsilon\)-uniform is given in Chapter 6. The next two chapters are concerned with linear convection-diffusion problems in one dimension. Chapter 7 contains technical results about upwind finite difference operators on fitted meshes, which are required for the proof in Chapter 8 that such fitted mesh methods are \(\varepsilon\)-uniform for these problems. In Chapter 9 finite element methods on fitted meshes for linear convection-diffusion problems in one dimension are constructed and a proof that they are \(\varepsilon\)-uniform is given. The use of the Schwarz iterative method is illustrated in Chapter 10, where it is applied to the one-dimensional linear convection-diffusion equation. A proof that the method is \(\varepsilon\)-uniform is also presented.

The remainder of the book is devoted to problems in two dimensions. Several linear convection-diffusion problems in two dimensions, and their numerical solution, are described in Chapter 11. In Chapter 12 bounds are obtained for derivations of the solutions of such problems in the case where only regular layers occur. Then in Chapter 13 these bounds are used to establish the fact that the fitted mesh method constructed for these problems in Chapter 11 is \(\varepsilon\)-uniform. Chapter 14 contains the surprising result that it is impossible to construct an \(\varepsilon\)-uniform numerical method using a fitted operator method on uniform rectangular meshes for problems with parabolic boundary layers. It is also indicated that, for such problems, \(\varepsilon\)-uniform fitted mesh methods are quite easy to construct. Finally in Chapter 15 it is proved that it is impossible to construct an \(\varepsilon\)-uniform numerical method using a standard finite difference operator on a fitted rectangular mesh for a problem having both an initial and a parabolic boundary layer. It is also indicated that for such problems \(\varepsilon\)-uniform numerical methods can be constructed using both a finite difference operator and a piecewise uniform fitted mesh.

The book ends with an Appendix, which contains a brief review of some classical bounds on derivatives of solutions of partial differential equations, stated in the terminology used in this book.

This book falls naturally into three parts. The first three chapters provide motivation and an elementary introduction to some aspects of the subject. The next seven chapters are concerned with problems exclusively in one dimension, while in the final five chapters problems in two spatial dimensions or in one spatial dimension and time are considered.

In the first three chapters simple examples of various one-dimensional problems involving singular perturbations are described and some issues concerning their numerical solution are discussed. The fact that such problems cannot be solved numerically in a completely satisfactory manner by standard methods is then explained. This indicates the need for methods that behave uniformly well whatever the value of the singular perturbation parameter. Such methods are called \(\varepsilon\)-uniform methods, where \(\varepsilon\) is the singular perturbation parameter.

Simple examples of \(\varepsilon\)-uniform finite difference methods are presented in Chapters 4 and 5. These are of two kinds: the first are the fitted operator methods which comprise specially designed finite difference operators on standard meshes; the second are the fitted mesh methods which comprise standard finite difference operators on specially designed meshes.

In Chapter 4 fitted operator methods on uniform meshes are described for some simple problems in one dimension. The chapter concludes with the construction of the EL-Mistikawy-Werle fitted operator method for linear convection-diffusion equations in one dimension and with a modern proof that it is an \(\varepsilon\)-uniform method. In Chapter 5 fitted mesh methods for simple problems in one dimension are constructed. A simple basic lemma is established and it is then proved that fitted mesh methods for initial value problems are \(\varepsilon\)-uniform. The proof that fitted mesh methods for linear reaction-diffusion equations in one dimension are \(\varepsilon\)-uniform is given in Chapter 6. The next two chapters are concerned with linear convection-diffusion problems in one dimension. Chapter 7 contains technical results about upwind finite difference operators on fitted meshes, which are required for the proof in Chapter 8 that such fitted mesh methods are \(\varepsilon\)-uniform for these problems. In Chapter 9 finite element methods on fitted meshes for linear convection-diffusion problems in one dimension are constructed and a proof that they are \(\varepsilon\)-uniform is given. The use of the Schwarz iterative method is illustrated in Chapter 10, where it is applied to the one-dimensional linear convection-diffusion equation. A proof that the method is \(\varepsilon\)-uniform is also presented.

The remainder of the book is devoted to problems in two dimensions. Several linear convection-diffusion problems in two dimensions, and their numerical solution, are described in Chapter 11. In Chapter 12 bounds are obtained for derivations of the solutions of such problems in the case where only regular layers occur. Then in Chapter 13 these bounds are used to establish the fact that the fitted mesh method constructed for these problems in Chapter 11 is \(\varepsilon\)-uniform. Chapter 14 contains the surprising result that it is impossible to construct an \(\varepsilon\)-uniform numerical method using a fitted operator method on uniform rectangular meshes for problems with parabolic boundary layers. It is also indicated that, for such problems, \(\varepsilon\)-uniform fitted mesh methods are quite easy to construct. Finally in Chapter 15 it is proved that it is impossible to construct an \(\varepsilon\)-uniform numerical method using a standard finite difference operator on a fitted rectangular mesh for a problem having both an initial and a parabolic boundary layer. It is also indicated that for such problems \(\varepsilon\)-uniform numerical methods can be constructed using both a finite difference operator and a piecewise uniform fitted mesh.

The book ends with an Appendix, which contains a brief review of some classical bounds on derivatives of solutions of partial differential equations, stated in the terminology used in this book.

Reviewer: H.Marcinkowska (Wrocław)

##### MSC:

65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |

65F10 | Iterative numerical methods for linear systems |

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |

35B25 | Singular perturbations in context of PDEs |

35K15 | Initial value problems for second-order parabolic equations |

65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |

65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |