Recipes for efficient higher-order multiscale asymptotic analysis

Realistic models for physical phenomena in singular perturbation theory often involve multiscale processes which need to be regularized systematically and carefully. A case in point is when the roughness (non-smoothness) of the solution is confined to a small portion of the simulation domain. In the past decades considerable efforts have been made in developing asymptotic solutions for multiscale problems. However, the majority of works only provide the first few dominant expansion terms due to the complexity of the iteration involved in the so-called ldquoMethod of Matched Asymptotic Expansions.rdquo The primary objectives in this paper are (i) to establish a convenient symbolic notation for the derivation of the asymptotics to any order desired; (ii) to demonstrate that the involved matching process in the algorithms occurs on the boundary rather than on boundary layers; (iii) to propose a technique which revises the usage of the O - calculus by making redundant a variety of intermediate calculation steps in standard analyses. Details will be discussed in terms of a model problem.