Abstract

Researchers in the Singular-Perturbation-Problems (SPP) community have largely focused on developing codes to solve problems with exponential layers.   Such codes, while generally effective in solving the problems for which they were designed, are difficult or impossible to adapt to problems with non-exponential layers, which are incomparably broader than exponential layers. The purpose of this paper is to demonstrate the need for increased efforts to study non-exponential layers and to develop appropriate new approaches to solving problems involving such layers, in particular new grid-clustering algorithms that rely on the layer structure to solve both model and practical problems.  The paper reviews recent achievements in this area.  It summarises the current state of knowledge on layer structures, including descriptions of the types of layers discovered so far, with emphasis on non-exponential ones. The paper outlines model problems involving such layers and the conditions under which they occur, describes specific techniques for generating layer-eliminating coordinate transformations and layer-resolving grids, outlines methods for solving certain model problems, and discusses possible applications of the methods to the numerical analysis of practical problems involving layers. It illustrates the theoretical and numerical methodology for a specific problem involving non-exponential layers.

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