Applications of Singularity Perturbed Differential Equations: Some Numerical Treatment

Advances in Numerical Treatment of Singularly Perturbed Differential Equations

Abstract

Singular perturbation problems (SPPs) arise frequently in fluid dynamics, quantum mechanics, chemical reactor theory and several other branches of applied mathematics. A wide variety of methods such as perturbation methods, Pad6 approximation methods and numerical methods are available in the literature to solve these problems. The numerical treatment of SPPs has received a significant amount of attention in recent years. It is a well-known fact that the solutions of SPPs exhibit a multiscale character.In this talk, I will discuss the role of numerical analysis in the design of numerical al¬gorithms to approximately solve certain classes of singularly perturbed differential equations. The solutions of singularly perturbed differential equation have narrow layer regions in the domain, where the solution exhibits steep gradients. Classical numerical methods suffer major defects in these regions. Alternative computational approaches will be discussed and the central issues in the associated numerical analy¬sis of these layer-adapted algorithms will be outlined.We present new results in the numerical analysis of singularly perturbed convection-diffusion- reaction problems that have appeared in the last five years. Mainly discussing layer-adapted meshes, we present also a survey on stabilization methods, adaptive methods, and on systems of singularly perturbed equations.