pt. 1. Ordinary differential equations. -- The analytical behaviour of solutions -- Numerical methods for second-order boundary value problems -- pt. 2. Parabolic initial-boundary value problems in one space dimension. -- Introduction -- Analytical behaviour of solutions -- Finite difference methods -- Finite element methods -- Two adaptive methods -- pt. 3. Elliptic and parabolic problems in several space dimensions. -- Analytical behaviour of solutions -- Finite difference methods -- Finite element methods -- Time-dependent problems -- pt. 4. The incompressible Navier-Stokes equations. -- Existence and uniqueness results -- Upwind finite element method -- Higher-order methods of streamline diffusion type -- Local projection stabilization for equal-order interpolation -- Local projection method for Inf-Sup stable elements -- Mass conservation for coupled flow-transport problems -- Adaptive error control.
This considerably expanded and completely revised second edition incorporates many new developments in the thriving field of numerical methods for singularly perturbed differential equations. It provides a thorough foundation for the numerical analysis and solution of these problems, which model many physical phenomena whose solutions exhibit layers. The book focuses on linear convection-diffusion equations and on nonlinear flow problems that appear in computational fluid dynamics. It offers a comprehensive overview of suitable numerical methods while emphasizing those with realistic error estimates. The book should be useful for scientists requiring effective numerical methods for singularly perturbed differential equations.
9783540344667
Singular perturbations (Mathematics) Numerical analysis Navier-Stokes equations Boundary value problems Finite element method Differential equations--Numerical solutions Mathematics Statistics