Numerical Analysis of Singular Perturbations and Convection-Diffusion Equations

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This project is concerned with the numerical analysis of singularly perturbed ordinary and partial differential equations. Standard numerical approaches have difficulty in such situations. The main objective of this project is to develop tools for the construction and analysis of high order finite-difference discretizations for these problems. The methods employed will consist of a combination of classical and modern notions of consistency, stability, and regularity. The development of the discretization schemes will use a general finite-difference framework and will take place in an environment of realistic numerical experimentation and rigorous mathematical analysis.