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2016 | Buch

Mathematical Modeling and PDEs Kapitel lesen Erstes Kapitel lesen

Partial Differential Equations: Modeling, Analysis and Numerical Approximation

verfasst von: Hervé Le Dret, Brigitte Lucquin

Verlag: Springer International Publishing

Buchreihe : ISNM International Series of Numerical Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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SUCHEN

Über dieses Buch

This book is devoted to the study of partial differential equation problems both from the theoretical and numerical points of view. After presenting modeling aspects, it develops the theoretical analysis of partial differential equation problems for the three main classes of partial differential equations: elliptic, parabolic and hyperbolic. Several numerical approximation methods adapted to each of these examples are analyzed: finite difference, finite element and finite volumes methods, and they are illustrated using numerical simulation results. Although parts of the book are accessible to Bachelor students in mathematics or engineering, it is primarily aimed at Masters students in applied mathematics or computational engineering. The emphasis is on mathematical detail and rigor for the analysis of both continuous and discrete problems.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Mathematical Modeling and PDEs
Abstract
In this chapter, we consider several concrete situations stemming from various areas of applications, the mathematical modeling of which involves partial differential equation problems. We will not be rigorous mathematically speaking. There will be quite a few rather brutal approximations, not always convincingly justified. This is however the price to be paid if we want to be able to derive mathematical models that aim to describe the complex phenomena we are dealing with in a way that remains manageable. At a later stage, we will study some of these models with all required mathematical rigor. The simplest examples arise in mechanics. Let us start with the simplest example of all.
Hervé Le Dret, Brigitte Lucquin
Chapter 2. The Finite Difference Method for Elliptic Problems
Abstract
Even though it can be shown that the boundary value problems introduced in Chap. 1 admit solutions, they do not admit explicit solutions as a general rule, i.e., solutions that can be written in closed form. Therefore, in order to obtain quantitative information on the solutions, it is necessary to define approximation procedures that are effectively computable. We present in this chapter the simplest of all approximation methods, the finite difference method. We apply the method to the numerical approximation of a model problem, the Dirichlet problem for the Laplacian in one space dimension. We then give some indications for the extension of the method to Neumann boundary conditions and to two-dimensional problems.
Hervé Le Dret, Brigitte Lucquin
Chapter 3. A Review of Analysis
Abstract
In order to go beyond the somewhat naive existence theory and finite difference method of approximation of elliptic boundary value problems seen in Chaps. 1 and 2, we need to develop a more sophisticated point of view. This requires in turn some elements of analysis pertaining to function spaces in several variables, starting with some abstract Hilbert space theory. This is the main object of this chapter. As already mentioned in the preface, this chapter can be read quickly at first, for readers who are not too interested in the mathematical details and constructions therein. A summary of the important results needed for the subsequent chapters is thus provided at the end of the chapter.
Hervé Le Dret, Brigitte Lucquin
Chapter 4. The Variational Formulation of Elliptic PDEs
Abstract
We now begin the theoretical study of elliptic boundary value problems in a context that is more general than the one-dimensional model problem treated in Chap. 1. We will focus on one approach, which is called the variational approach. There are other ways of solving elliptic problems, such as working with Green functions as seen in Chap. 2. The variational approach is quite simple and well suited for a whole class of approximation methods, as we will see later.
Hervé Le Dret, Brigitte Lucquin
Chapter 5. Variational Approximation Methods for Elliptic PDEs
Abstract
One of the virtues of the variational approach is that it leads naturally to a whole family of approximation methods. Let us emphasize again that the reason why approximation methods for PDEs are needed is that, even though we may be able to prove the existence of a solution, in general there is no closed form formula for it. There are other approximation methods that are not variational, such as the finite difference method seen earlier, the finite volume method that we will see in Chap. 10, and yet many other methods that we will not consider in this book.
Hervé Le Dret, Brigitte Lucquin
Chapter 6. The Finite Element Method in Dimension Two
Abstract
It should already be clear that there is no difference between elliptic problems in one dimension and elliptic problems in several dimensions from the variational viewpoint. The same goes for the abstract part of variational approximations. The difference lies in the description of the finite dimensional approximation spaces. The FEM in any dimension of space is based on the same principle as in one dimension, that is to say, we consider spaces of piecewise polynomials of low degree, with lots of pieces for accuracy. Now things are right away quite different, and actually considerably more complicated, since polynomials have several variables, and open sets are much more varied than in dimension one. For simplicity, we limit ourselves to the two-dimensional case.
Hervé Le Dret, Brigitte Lucquin
Chapter 7. The Heat Equation
Abstract
We have so far studied elliptic problems, i.e., stationary problems. We now turn to evolution problems, starting with the archetypal parabolic equation, namely the heat equation. In this chapter, we will present a brief and far from exhaustive theoretical study of the heat equation. We will mostly work in one dimension of space, some of the results having an immediate counterpart in higher dimensions, others not. The study of numerical approximations of the heat equation will be the subject of the next chapter.
Hervé Le Dret, Brigitte Lucquin
Chapter 8. The Finite Difference Method for the Heat Equation
Abstract
We now turn to numerical methods that can be used to approximate the solution of the heat equation. We develop the finite difference method in great detail, with particular emphasis on stability issues, which are delicate. We concentrate on the heat equation in one dimension of space, with homogeneous Dirichlet boundary conditions. We also give some indications about finite difference (in time)-finite element (in space) approximation.
Hervé Le Dret, Brigitte Lucquin
Chapter 9. The Wave Equation
Abstract
In this chapter, we present a short and even more far from exhaustive theoretical study of the wave equation. We establish the existence and uniqueness of the solution, as well as the energy estimates. We describe the qualitative behavior of solutions, which is very different from that of the heat equation. Again, we will mostly work in one dimension of space. In the same chapter, we introduce finite difference methods for the numerical approximation of the wave equation. Here again, stability issues are prominent, and significantly more delicate than for the heat equation.
Hervé Le Dret, Brigitte Lucquin
Chapter 10. The Finite Volume Method
Abstract
The finite volume method is a more recent method than both finite difference and finite element methods. It is widely used in practice for example in fluid dynamics computations. We present the method in the simplest possible settings, first for one-dimensional elliptic problems, then for the transport equation in one and two dimensions. For the latter, we return to the method of characteristics already introduced in Chap. 1, to solve one-dimensional nonlinear problems as well as two-dimensional linear problems.
Hervé Le Dret, Brigitte Lucquin
Backmatter
Metadaten
Titel
Partial Differential Equations: Modeling, Analysis and Numerical Approximation
verfasst von
Hervé Le Dret
Brigitte Lucquin
Copyright-Jahr
2016
Electronic ISBN
978-3-319-27067-8
Print ISBN
978-3-319-27065-4
DOI
https://doi.org/10.1007/978-3-319-27067-8