Numerical Approximation of Partial Differential Equations

1. Front Matter

   Pages I-XVI

   PDF

2. Basic Concepts and Methods for PDEs’ Approximation

   1. Introduction

      Pages 1-16

   2. Numerical Solution of Linear Systems

      Pages 17-71

   3. Finite Element Approximation

      Pages 73-100

   4. Polynomial Approximation

      Pages 101-127

   5. Galerkin, Collocation and Other Methods

      Pages 129-157

3. Approximation of Boundary Value Problems

   1. Elliptic Problems: Approximation by Galerkin and Collocation Methods

      Pages 159-216

   2. Elliptic Problems: Approximation by Mixed and Hybrid Methods

      Pages 217-255

   3. Steady Advection-Diffusion Problems

      Pages 257-296

   4. The Stokes Problem

      Pages 297-337

   5. The Steady Navier-Stokes Problem

      Pages 339-362

4. Approximation of Initial-Boundary Value Problems

   1. Parabolic Problems

      Pages 363-404

   2. Unsteady Advection-Diffusion Problems

      Pages 405-427

   3. The Unsteady Navier-Stokes Problem

      Pages 429-448

   4. Hyperbolic Problems

      Pages 449-508

5. Back Matter

   Pages 509-543

   PDF

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).

• Algorithmen
• Approximation
• Numerische Approximation
• algorithm
• algorithms
• differential equation
• finite Elemente
• finite element method
• finite elements
• numerical methods
• partial differential equation
• partial differential equations
• partielle Differentialgleichung
• partielle Differentialgleichungen
• spektral Methoden

"...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

• École Polytechnique Fédérale de Lausanne Chaire de Modelisation et Calcul Scientifique (CMCS), Lausanne, Switzerland

  Alfio Quarteroni

• Politecnico di Milano, MOX, Milan, Italy

  Alfio Quarteroni

• Dipartimento di Matematica, Università di Trento, Povo TN, Italy

  Alberto Valli

Policies and ethics