This book deals with the numerical approximation of partial differential equations. Its scope is to provide a thorough illustration of numerical methods, carry out their stability and convergence analysis, derive error bounds, and discuss the algorithmic aspects relative to their implementation. A sound balancing of theoretical analysis, description of algorithms and discussion of applications is one of its main features. Many kinds of problems are addressed. A comprehensive theory of Galerkin method and its variants, as well as that of collocation methods, are developed for the spatial discretization. These theories are then specified to two numerical subspace realizations of remarkable interest: the finite element method and the spectral method. From the reviews: "...The book is excellent and is addressed to post-graduate students, research workers in applied sciences as well as to specialists in numerical mathematics solving PDE. Since it is written very clearly, it would be acceptable for undergraduate students in advanced courses of numerical mathematics. Readers will find this book to be a great pleasure."--MATHEMATICAL REVIEWS

Klappentext

Everything is more simple than one thinks but at the same time more complex than one can understand Johann Wolfgang von Goethe To reach the point that is unknown to you, you must take the road that is unknown to you St. John of the Cross This is a book on the numerical approximation ofpartial differential equations (PDEs). Its scope is to provide a thorough illustration of numerical methods (especially those stemming from the variational formulation of PDEs), carry out their stability and convergence analysis, derive error bounds, and discuss the algorithmic aspects relative to their implementation. A sound balancing of theoretical analysis, description of algorithms and discussion of applications is our primary concern. Many kinds of problems are addressed: linear and nonlinear, steady and time-dependent, having either smooth or non-smooth solutions. Besides model equations, we consider a number of (initial-) boundary value problems of interest in several fields of applications. Part I is devoted to the description and analysis of general numerical methods for the discretization of partial differential equations. A comprehensive theory of Galerkin methods and its variants (Petrov­ Galerkin and generalized Galerkin), as wellas ofcollocationmethods, is devel­ oped for the spatial discretization. This theory is then specified to two numer­ ical subspace realizations of remarkable interest: the finite element method (conforming, non-conforming, mixed, hybrid) and the spectral method (Leg­ endre and Chebyshev expansion).

Inhalt

Basic Concepts and Methods for PDEs' Approximation.- Numerical Solution of Linear Systems.- Finite Element Approximation.- Polynomial Approximation.- Galerkin, Collocation and Other Methods.- Approximation of Boundary Value Problems.- Elliptic Problems: Approximation by Galerkin and Collocation Methods.- Elliptic Problems: Approximation by Mixed and Hybrid Methods.- Steady Advection-Diffusion Problems.- The Stokes Problem.- The Steady Navier-Stokes Problem.- Approximation of Initial-Boundary Value Problems.- Parabolic Problems.- Unsteady Advection-Diffusion Problems.- The Unsteady Navier-Stokes Problem.- Hyperbolic Problems.