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1994 | Buch

Introduction Kapitel lesen Erstes Kapitel lesen

Numerical Approximation of Partial Differential Equations

verfasst von: Alfio Quarteroni, Alberto Valli

Verlag: Springer Berlin Heidelberg

Buchreihe : Springer Series in Computational Mathematics

Enthalten in: Professional Book Archive

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SUCHEN

Über dieses Buch

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).

Inhaltsverzeichnis

Frontmatter

Basic Concepts and Methods for PDEs’ Approximation

1.. Introduction
Abstract
Numerical approximation of partial differential equations is an important branch of Numerical Analysis. Often, it demands a knowledge of many aspects of the problem.
2.. Numerical Solution of Linear Systems
Abstract
The solution of linear algebraic systems lies at the heart of most calculations in scientific computing. Here we describe some of the most populat methods that are applied to systems of general form. For a more complete presentation we refer the reader to the literature that we will quote throughout this Chapter. Special techniques for systems arising from the discretization of partial differential equations are discussed in this book for each specific situation.
3.. Finite Element Approximation
Abstract
In this Chapter we present the properties of the classical finite element approximation. We underline the three basic aspects of this method: the existence of a triangulation of ω, the construction of a finite dimensional subspace consisting of piecewise-polynomials, and the existence of a basis of functions having small support. Then, we introduce the interpolation operator and we estimate the interpolation error. Some final remarks will be devoted to several projection operators upon finite element subspaces and their approximation properties.
4.. Polynomial Approximation
Abstract
This Chapter is devoted to the introduction of basic notions and working tools concerning orthogonal algebraic polynomials. More specifically, we will present some properties of both Chebyshev and Legendre polynomials, concerning projection and interpolation processes. These will provide the background of spectral methods for the approximation of partial differential equations that are considered throughout Part II and III of this book.
5.. Galerkin, Collocation and Other Methods
Abstract
This Chapter is devoted to a short presentation of some classical techniques for the discretization of (initial-) boundary value problems.

Approximation of Boundary Value Problems

6.. Elliptic Problems: Approximation by Galerkin and Collocation Methods
Abstract
This Chapter deals with second order linear elliptic equations. We present the variational formulation of some classical boundary value problems, accounting for several kind of boundary conditions, and derive existence and uniqueness of the solution. Then we approximate these problems by Galerkin and collocation methods, in the framework of both finite element and spectral methods. For each kind of discretization we analyze its stability and convergence properties, as well as its algorithmic aspects.
7.. Elliptic Problems: Approximation by Mixed and Hybrid Methods
Abstract
In this Chapter, we consider some alternative formulations of second order linear elliptic equations, based on variational principles that differ from the classical one presented in the previous Chapter. We then introduce new numerical approaches that are tailored on the new formulations. More precisely, we first introduce the so-called equilibrium finite element methods through the minimization of the complementary energy. We then consider the Hellinger-Reissner principle, giving rise to both mixed and hybrid finite element methods. Finally, a more general analysis of saddle-point problems and their approximation via Lagrangian multipliers is provided.
8.. Steady Advection-Diffusion Problems
Abstract
In this Chapter we consider second order linear elliptic equations that have an advective term which is much stronger than the diffusive one.
9.. The Stokes Problem
Abstract
This Chapter is devoted to the Stokes system of differential equations, that can be written as
$$ \left\{ {\begin{array}{*{20}c} {a_0 u - \nu \Delta u + \nabla p = f in \Omega \subset \mathbb{R}^d , d = 2,3} \\ {div u = 0 in \Omega } \\ {u = 0 on \partial \Omega , } \\ \end{array} } \right. $$
(9.1)
where a0≥0 and ν>0 are given constants, f: Ω → ℝd is a given function, while u: Ω → ℝd and p: Ω → ℝ are the problem unknowns.
10.. The Steady Navier-Stokes Problem
Abstract
This Chapter addresses the steady Navier-Stokes equations, which describe the motion, independent of time, of a homogeneous incompressible fluid. A complete derivation of these equations will be provided in Section 13.1.

Approximation of Initial-Boundary Value Problems

11.. Parabolic Problems
Abstract
This Chapter deals with linear second-order parabolic equations. The typical example is given by the heat equation, which is the non-stationary counterpart of the Laplace equation.
12.. Unsteady Advection-Diffusion Problems
Abstract
In this Chapter we consider linear second-order parabolic equations with advective terms dominating over the diffusive ones. The corresponding steady case was considered in Chapter 8.
13.. The Unsteady Navier-Stokes Problem
Abstract
In this Chapter we turn our attention towards unsteady viscous flows, especially in the incompressible case. We therefore consider the time-dependent counterpart of the Navier-Stokes problem (10.1.1)-(10.1.3), that reads
$$ \left\{ {\begin{array}{*{20}c} {\frac{{\partial u}} {{\partial t}} - \nu \Delta u + \left( {u \cdot \nabla } \right)u + \nabla p = f in Q_T : = \left( {0,T} \right) \times \Omega } \\ \begin{gathered} div u = 0 in Q_T \hfill \\ u = 0 on \Sigma _{\rm T} : = \left( {0,T} \right) \times \partial \Omega \hfill \\ \end{gathered} \\ {u_{|t = 0} = u_0 on \Omega , } \\ \end{array} } \right. $$
(13.1)
where f=f(t, x) and u0=u0(x) are given data, ω is an open bounded domain of ℝ d , with d=2,3, and ηω is its boundary. One remarkable feature of (13.1) is the absence of an equation containing ηp/ηt. Indeed, in (13.1) the pressure p appears as a Lagrange multiplier associated to the divergence-free constraint div u=0.
14.. Hyperbolic Problems
Abstract
The numeical approximation of hyperbolic equations is a v ery active area of reasearch. The main distinguishing feature of these initial-boundary value problems is the fact that perturbations propagate with finite speed. Another characterizing aspect is that the boundary treatment is not as simple as that for elliptic or parabolic equations. According to the sign of the equation coefficients, the inflow and outflow boundary regions determine, from case to case, where boundary conditions have to be prescribed. The situation becomes more complex for systems of hyperbolic equations, where the boundary treatment must undergo a local characteristic analysis. If not implemented conveniently, the numerical realization of boundary conditions is a potential source of spurious instabilities.
Backmatter
Metadaten
Titel
Numerical Approximation of Partial Differential Equations
verfasst von
Alfio Quarteroni
Alberto Valli
Copyright-Jahr
1994
Verlag
Springer Berlin Heidelberg
Electronic ISBN
978-3-540-85268-1
Print ISBN
978-3-540-85267-4
DOI
https://doi.org/10.1007/978-3-540-85268-1