Kolmogorov N-widths are an approximation theory concept that, for a given problem, yields information about the optimal rate of convergence attainable by any numerical method applied to that problem. We survey sharp bounds recently obtained for the N-widths of certain singularly perturbed convection-diffusion and reaction-diffusion boundary value problems.

The singularly perturbed boundary blow-up problem −ε2∆u = u(u− a)(1− u) u > 0 in B, u = ∞ on ∂B is studied in the unit ball B ⊂ R (N ≥ 2), a ∈ (1/2, 1) is a constant. It is shown that there exist exactly three positive solutions for the problem and all of them are radially symmetric solutions.

This paper is concerned with the upwind finite-difference discretization of a quasilinear singularly perturbed boundary value problem without turning points. Kopteva’s a posteriori error estimate [N. Kopteva, Maximum norm a posteriori error estimates for a onedimensional convection-diffusion problem, SIAM J. Numer. Anal., 39, 423–441 (2001)] is generalized and improved. 2000 MSC: 65L10, 65L70.

The linear singularly perturbed reaction-diffusion problem is considered. The spline difference scheme on the Shishkin mesh is used to solve the problem numerically. With the special position of collocation points, the obtained scheme satisfies the discrete minimum principle. Numerical experiments which confirm theoretical results are presented. AMS Mathematics Subject Classification (2000): 65...

The aim of this study is to develop a regularization method for boundary value problems parabolic equation. A singularly perturbed problem on the semiaxis considered in case “simple” rational turning point. To prove asymptotic convergence series, maximum principle used.

Foreword 2

Nonlinear singularly perturbed interior layer problems are examined. Numerical results are presented for a numerical method consisting of a monotone scheme on a Shishkin mesh refined around the approximate location of the interior layer. keywords: Singular Perturbation, Shishkin mesh, Nonlinear, Interior Layer

Problem (1.1) appears in applied mathematics. See for example [13, 14] and the references therein. For the interesting link between this problem and the modelling of activator-inhibitor systems, the authors can refer to [11]. In [13, 14], Ni and Takagi prove that the least energy solution of (1.1) has exactly one local maximum point xε which lies in ∂Ω, and xε tends to a point x0 which attains ...

We consider the following singularly perturbed Neumann problem ε∆u− u + u = 0 , u > 0 in Ω, ∂u ∂ν = 0 on ∂Ω, where p > 1 and Ω is a smooth and bounded domain in R. We construct a class of solutions which consist of large number of spikes concentrating on three line segments with a common endpoint which intersect ∂Ω orthogonally .

A semilinear singularly perturbed reaction-diffusion problem is considered and the approximate solution is given in the form of a quadratic polynomial spline. Using the collocation method on a simple piecewise equidistant mesh, an approximation almost second order uniformly accurate in small parameter is obtained. Numerical results are presented in support of this result. AMS Mathematics Subjec...