Numerical Methods for Singularly Perturbed Differential Equations

1. Front Matter

   Pages I-XVI

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2. Ordinary Differential Equations

   • Hans-Görg Roos, Martin Stynes, Lutz Tobiska

   Pages 1-103

3. Parabolic Initial-Boundary Value Problems in One Space Dimension

   • Hans-Görg Roos, Martin Stynes, Lutz Tobiska

   Pages 105-171

4. Elliptic Boundary Value Problems

   • Hans-Görg Roos, Martin Stynes, Lutz Tobiska

   Pages 173-278

5. Incompressible Navier-Stokes Equations

   • Hans-Görg Roos, Martin Stynes, Lutz Tobiska

   Pages 279-320

6. Back Matter

   Pages 321-351

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The analysis of singular perturbed differential equations began early in this century, when approximate solutions were constructed from asymptotic ex­ pansions. (Preliminary attempts appear in the nineteenth century [vD94].) This technique has flourished since the mid-1960s. Its principal ideas and methods are described in several textbooks. Nevertheless, asymptotic ex­ pansions may be impossible to construct or may fail to simplify the given problem; then numerical approximations are often the only option. The systematic study of numerical methods for singular perturbation problems started somewhat later - in the 1970s. While the research frontier has been steadily pushed back, the exposition of new developments in the analysis of numerical methods has been neglected. Perhaps the only example of a textbook that concentrates on this analysis is [DMS80], which collects various results for ordinary differential equations, but many methods and techniques that are relevant today (especially for partial differential equa­ tions) were developed after 1980.Thus contemporary researchers must comb the literature to acquaint themselves with earlier work. Our purposes in writing this introductory book are twofold. First, we aim to present a structured account of recent ideas in the numerical analysis of singularly perturbed differential equations. Second, this important area has many open problems and we hope that our book will stimulate further investigations.Our choice of topics is inevitably personal and reflects our own main interests.

• Boundary value problem
• Grenzschichten
• Navier-Stokes-Gleichungen
• Navier-stokes equations
• Singular perturbation
• Singuläre Störungen
• convection-diffusion
• differential equation
• finite element method
• model
• numerical analysis
• numerical methods
• numerische Methoden
• ordinary differential equation
• partial differential equation

• Institut für Numerische Mathematik, Technische Universität Dresden, Dresden, Germany

  Hans-Görg Roos

• Department of Mathematics, University College Cork, Cork, Ireland

  Martin Stynes

• Institut für Analysis und Numerik, Otto-von-Guericke Universität Magdeburg, Magdeburg, Germany

  Lutz Tobiska

Policies and ethics