Beginning with ordinary differential equations, then moving on to parabolic and elliptic problems and culminating with the Navier-Stokes equations, the reader is led through the theoretical and practical aspects of the most important methods used to compute numerical solutions for singular perturbation problems.
The analysis of singular perturbed differential equations began early in the century with approximate solutions derived from asymptotic expansions, a technique that gained traction in the mid-1960s. While several textbooks cover its principal ideas and methods, there are instances where asymptotic expansions may be infeasible or fail to simplify the problem, making numerical approximations the only viable alternative. The systematic study of numerical methods for these singular perturbation problems emerged in the 1970s. Despite ongoing advancements, the exposition of new developments in numerical methods has been overlooked, with the notable exception of a textbook that compiles results for ordinary differential equations. Many relevant methods for partial differential equations have emerged since 1980, leaving contemporary researchers to sift through literature to find earlier work. This introductory book aims to provide a structured overview of recent ideas in the numerical analysis of singularly perturbed differential equations while highlighting the numerous open problems in the field, thereby encouraging further exploration. The topics selected reflect the authors' personal interests and expertise.
KlappentextDieses Lehrbuch ist eine verständlich geschriebene, kompakte Einführung in die numerische Mathematik. Es wendet sich an all jene, die numerische Verfahren zur Computersimulation realer Prozesse mittels mathematischer Modelle einsetzen und die Grundgedanken der dazu geeigneten Verfahren verstehen wollen. Schwerpunkte bilden numerische Verfahren für lineare und nichtlineare Gleichungssysteme, Eigenwertaufgaben, Interpolation und Approximation, numerische Differentiation und Integration sowie für Anfangswertaufgaben bei gewöhnlichen und Randwertaufgaben bei partiellen Differentialgleichungen.