Manuscript Topics

Singular perturbation problems are pervasive across a wide array of multidisciplinary fields, encompassing fluid dynamics (where they manifest in phenomena such as boundary layers and shock waves), quantum mechanics, materials science, and optimal control theory, among others. Such problems are characterized by the presence of small parameters that multiply the highest-order derivative(s) in the governing equations, giving rise to unique and challenging behaviors. These behaviors include the formation of sharp transition layers, rapid oscillations, and multiscale dynamics, which collectively pose significant hurdles for conventional numerical methods. These traditional approaches often suffer from computational instabilities, a notable loss of accuracy, and prohibitively high computational costs when attempting to resolve the intricate features inherent in singularly perturbed systems. Although numerical methods for singularly perturbed problems get a booming development in the past several decades, there still exist notable limitations and challenges in their application. Therefore, developing the numerical solutions of singularly perturbed problems is still quite challenging in the field of computational mathematics.    
The primary objective of this special issue is to spotlight some of the latest advancements in the development of efficient numerical methods tailored for singularly perturbed problems. We warmly invite submissions of articles and comprehensive reviews that delve into this theme, specifically focusing on numerical approaches to tackle singularly perturbed problems.

Topics include, but are not limited to the following:

Finite element method;
Discontinuous Galerkin method;
Finite difference method;
Spectral, hp-methods;
Error analysis and posteriori error estimate;
Nonuniform and adaptive discretizations;
Modeling and simulations involving singularly perturbed PDEs;
Shishkin/Bakhvalov-type grids, anisotropic refinements;
Real-world applications and computational aspects on fluid mechanics, semiconductor modeling, finance, and biophysics.

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