The wide use of computing techniques, combined with the demands of scientific and technical practices, has stimulated the development of numerical methods to a great extent, and in particular, methods for solving differential equations. The efficiency of such methods is governed by their accuracy, simplicity in computing the discrete solution and also their relative insensitivity to parameters in the problem. At present, numerical methods for solving partial differential equations, in particular, finite difference schemes, are well developed for wide classes of boundary value problems (see, for example, [79, 108, 100, 214, 91, 216]). Among boundary value problems, a considerable class includes problems for

singularly perturbed equations, i.e., differential equations whose highest-order derivatives are multiplied by a (perturbation) parameter ε. The perturbation parameter ε may take arbitrary values in the open-closed interval (0, 1] (see, e.g., [211, 210, 57, 94, 62]). Solutions of singularly perturbed problems, unlike regular problems, have boundary and/or interior layers, that is, narrow subdomains specified by the parameter ε on which the solutions vary by a finite value. The derivatives of the solution in these subdomains grow without bound as ε tends to zero. In the case of singularly perturbed problems, the use of numerical methods

developed for solving regular problems leads to errors in the solution that depend on the value of the parameter ε. Errors of the numerical solution depend on the distribution of mesh points and become small only when the effective mesh-size in the layer is much less than the value of the parameter ε (see, e.g., [138, 87, 106, 33]). Such numerical methods turn out to be inapplicable for singularly perturbed problems. Due to this, there is an interest in the development of special numerical

methods where solution errors are independent of the parameter ε and defined only by the number of nodes in the meshes used, i.e., numerical methods (in particular, finite difference schemes) that converge ε-uniformly. When the solutions by such methods are ε-uniformly convergent, we will call these methods and solutions robust (as in [33]). At present, only several books

perturbed problems. Grid methods for boundary value problems for partial differential equations are considered in the books [138, 87, 33, 75]; see also [26, 13, 14, 76] for ordinary differential equations. In the book [106], the authors give a number of results and also a comprehensive bibliography on numerical methods for solving singularly perturbed problems for partial differential equations and for ordinary differential equations. The present book was intended to be an English translation of the book

[138]. A variety of ideas and approaches from [138] have since been further developed. New approaches and trends appear, which require further investigation. In the present book, we elaborate on approaches to the development of ε-uniformly convergent numerical methods for several boundary value problems from [138] and discuss some new trends in the development of other methods, which have appeared recently. Quite often solutions of boundary value problems, their grid solutions, and

also their convergence are considered using maximum norms. The use of either the energy norm or L1, L2-norms is inadequate to describe the solutions of singularly perturbed problems and their approximations. For example, in the case of problems with a parabolic boundary layer, the boundary-layer function (that is finite in the maximum norm) tends to zero in the norms mentioned above as ε → 0 [87, 33]. In this book, maximum norms are consistently used. As a rule, we avoid reference to works where problems for singularly perturbed ordinary differential equations are considered since such results and techniques cannot, in general, be carried over to problems for partial differential equations. The first ε-uniformly convergent difference schemes constructed for singu-

larly perturbed problems used two main approaches: fitted operator method and condensing mesh (grid) method/fitted mesh (grid) method. Schemes based on the fitted operator method were constructed in [2] and, independently, constructed and justified in [56] (for ordinary differential equations); in [15], a scheme for the condensing mesh method was constructed and justified (for an elliptic equation). For schemes using condensing meshes, ε-uniform convergence of the solution of a difference scheme to the solution of the boundary value problem is guaranteed by a special choice of the distribution of mesh points (for the given number of nodes). Restrictions on the choice of difference equations approximating singularly perturbed problems (for ensuring the εuniform convergence of the scheme) are, in general, not imposed. In fitted operator methods, ε-uniform convergence of the solution of the difference scheme is achieved by a special choice of coefficients of the difference equations approximating the differential problem. Restrictions on the distribution of mesh points for ensuring the ε-uniform convergence of the scheme are not imposed. We mention also an approach related to additive splitting of a singularity

suggested for singularly perturbed problems in [12] (see also [11]). In this method, basic functions include special functions approximating the singular

problems for partial differential equations, this approach was not widely used because the singular components of solutions of boundary value problems have the form too complicated for the effective construction of a system of basis functions. The difference schemes based on the method of additive splitting of a singularity and constructed in [12, 11] converge ε-uniformly in the energy norm. After the publications [15, 56], there was a large effort to develop fitted op-

erator methods. The first book [26] is completely devoted to the development of such methods for ordinary differential equations. Later, fitted operator methods continued to be intensively developed (see, for example, a series of variants of the fitted operator schemes for elliptic equations in [103]). A comprehensive bibliography on numerical methods for singularly perturbed problems is given in [106]. After [15, 56], in the case of partial differential equations, the first finite difference schemes that converge ε-uniformly in the maximum norm are constructed in [29] (see also [30, 1] for the fitted operator scheme, and [74, 118] for the schemes on condensing meshes). Note that fitted operator methods (see their description, e.g., in [26, 87,

33, 106]) have an advantage in simplicity because meshes used are uniform, and this contributed to their more rapid progress compared with condensing mesh methods. However, fitted operator methods have a restricted domain of applicability for constructing ε-uniformly convergent numerical methods. It was first established in [124] that there are no ε-uniformly convergent

schemes based on the fitted operator method in the case for singularly perturbed elliptic convection-diffusion equations in domains where parts of the boundary are characteristics of a reduced equation and parabolic boundary layers appear. In the same paper, a scheme was constructed that converges ε-uniformly, using both the fitted operator method for the approximation of derivatives along characteristics of the reduced equation and the condensing mesh method for the approximation of derivatives in the direction orthogonal to the characteristics. The resulting discrete solutions also make it possible to approximate the normalized derivatives ε-uniformly. For parabolic equations with parabolic layers it is proved that there exist no

schemes using the fitted operator method that converge ε-uniformly in any of the papers: [130] in the case of a parabolic boundary layer and [148] in the case of a parabolic initial layer. When constructing schemes for nonlinear problems, the situation is much

more complicated. In [137] (for reaction-diffusion equations) and [139] (for convection-diffusion equations), it was established that even for semilinear ordinary differential equations there exist no schemes based on the fitted operator method that converge ε-uniformly. Similar difficulties related to the use of fitted operator methods in numerical methods are discussed in later publications (see, e.g., [138, 87, 86] for partial differential equations and [84, 32] for semilinear ordinary differential equations). Numerical experiments that

ordinary differential equation are considered in [33]. Thus, for a wide class of boundary value problems there are no schemes using fitted operator methods that converge ε-uniformly in the maximum norm; independently of these, schemes may be constructed using classical finite difference approximations or finite element or finite volume methods. Having summarizing the approaches to the construction of ε-uniformly con-

vergent numerical methods, we make one more remark. In works [134, 140] (see also discussions in [87, 83]), a class of problems is distinguished for elliptic and parabolic equations that degenerate on the boundary of the domain whose solutions contain initial and parabolic boundary layers. It is shown that for such problems there are no schemes of the condensing mesh method (for rectangular meshes) that converge ε-uniformly. But the application of both approaches-fitted mesh and fitted operator methods-makes it possible to construct schemes that converge ε-uniformly. Up to now, practically all singularly perturbed partial differential equations,

for which difference schemes that converge ε-uniformly in the maximum norm have been constructed, do not contain mixed derivatives. Boundary value problems have been considered only for the simplest subdomains of dimension not higher than two, and the elliptic operator in the differential equations is the Laplace operator. In the case of an elliptic equation with mixed derivatives considered only on a rectangle, the ε-uniformly convergent scheme obtained appears to be too complex and can not be extended to other dimensions in geometry [28]. Problems in domains with curvilinear boundaries are considered only in few publications. It is one of the goals of the present book to overcome such an existing

unsatisfactory state in the area of development of ε-uniformly convergent difference schemes. Another goal of the book is, on model problems, to consider some modern trends in the development of numerical methods for singularly perturbed problems that require further investigation.