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2015 | Buch

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Numerical Methods for Nonlinear Partial Differential Equations

verfasst von: Sören Bartels

Verlag: Springer International Publishing

Buchreihe : Springer Series in Computational Mathematics

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

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Über dieses Buch

The description of many interesting phenomena in science and engineering leads to infinite-dimensional minimization or evolution problems that define nonlinear partial differential equations. While the development and analysis of numerical methods for linear partial differential equations is nearly complete, only few results are available in the case of nonlinear equations. This monograph devises numerical methods for nonlinear model problems arising in the mathematical description of phase transitions, large bending problems, image processing, and inelastic material behavior. For each of these problems the underlying mathematical model is discussed, the essential analytical properties are explained, and the proposed numerical method is rigorously analyzed. The practicality of the algorithms is illustrated by means of short implementations.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract

Starting with the brachistochrone problem solved by Johann Bernoulli in the 17th century, differential equations have become an indispensable tool to model, understand, and solve real world problems.

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Analytical and Numerical Foundations

Frontmatter

Chapter 2. Analytical Background

Abstract

The calculus of variations provides an attractive mathematical framework for describing many phenomena in continuum mechanics. In this chapter model problems, including bending thin elastic objects as well as elastoplastic material behavior, are formulated within this framework, and general concepts such as the direct method in the calculus of variations and gradient flows that imply their well-posedness are discussed. The definitions and most important properties of Sobolev and Sobolev–Bochner spaces are provided.

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Chapter 3. FEM for Linear Problems

Abstract

The finite element method defines a general concept for systematically constructing finite-dimensional subspaces of Sobolev spaces that lead to discretizations of weak formulations of partial differential equations or minimization problems. The approximation properties of low-order finite element functions on simplicial meshes and their application to stationary and evolutionary linear model problems are discussed in this chapter. Short Matlab implementations illustrate the flexibility of the method and provide a tool to verify theoretical statements.

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Chapter 4. Concepts for Discretized Problems

Abstract

The rigorous justification of a numerical scheme for approximating a partial differential equation depends critically on the analytical properties of the problem and its solutions. If regularity and stability results are available, then error estimates can often be proved. Otherwise, weaker concepts, such as weak accumulation of approximations at exact solutions have to be employed. Discretizations of differential equations typically lead to nonlinear systems of equations, the convergence of iterative algorithms for their practical solution can only be guaranteed if the method respects the particular features of the underlying problem. In this chapter various techniques to prove convergence of discretizations and iterative algorithms are introduced, analyzed, and applied to model problems.

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Approximation of Classical Formulations

Frontmatter

Chapter 5. The Obstacle Problem

Abstract

The obstacle problem describes constrained deflection of a membrane and serves as a model problem for infinite-dimensional minimization problems with pointwise inequality constraint. Equivalent formulations of the problem including a variational inequality and a characterization with a Lagrange multiplier are derived. The well-posedness and convergence of lowest-order conforming finite element methods, together with a priori and a posteriori error estimates, are established. The application of the semismooth Newton method and a globally convergent primal-dual method together with their implementation conclude the chapter.

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Chapter 6. The Allen–Cahn Equation

Abstract

The mathematical description of phase separation and melting processes is often based on phase field models. These descriptions define a phase field variable that specifies particular phases as the solution of a semilinear parabolic partial differential equation. A small parameter that defines the width of the interfaces between different phases enters classical stability estimates in a critical way, and refined arguments are required to improve this dependence. Applying those estimates to analyzing approximation schemes for the simplest case of the Allen–Cahn equation in terms of a priori and a posteriori error estimates is carried out. The stability of various implicit and semi-implicit time-stepping schemes is discussed.

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Chapter 7. Harmonic Maps

Abstract

Partial differential equations with a nonlinear pointwise equality constraint arise in the mathematical modeling of liquid crystals and ferromagnets. The solutions are vector fields that describe the orientation of the molecules or the magnetization field. A simple model problem defines harmonic maps in the sphere as minimizers of the Dirichlet energy subject to a pointwise unit-length constraint. Lowest-order conforming finite element methods prohibit imposing the constraint everywhere and limited regularity and uniqueness properties of harmonic maps require a careful numerical treatment. The stability and convergence of iterative methods that employ linearizations and projections are proved, and short implementations are provided in this chapter. Extensions to approximating related evolution problems, such as the harmonic map heat flow and wave maps are discussed.

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Chapter 8. Bending Problems

Abstract

Deforming thin elastic objects within the bending regime is a classical problem in continuum mechanics and efficient mathematical descriptions lead to fourth-order problems. In the case of small displacements, a linear partial differential equation provides accurate approximations, but finite element methods have to be carefully developed to avoid locking phenomena. In the case of large deformations, a pointwise isometry constraint has to be incorporated which can be treated with techniques developed for approximating harmonic maps. Related problems arise in modeling of fluid membranes, but require different numerical methods. Iterative algorithms, together with short implementations, are motivated and analyzed.

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Methods for Extended Formulations

Frontmatter

Chapter 9. Nonconvexity and Microstructure

Abstract

Infinite-dimensional minimization problems without convexity properties may lead to the nonexistence of solutions, but arise as simplified mathematical descriptions of crystalline phase transitions that enable the shape-memory effect of smart materials. The ill-posed minimization problems capture important effects and relaxation theories define well-posed modifications of the functionals. The problems related to direct numerical treatment of the original formulations and the convergence and numerical solution of discretizations of its modifications are investigated in this chapter.

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Chapter 10. Free Discontinuities

Abstract

The mathematical description of problems involving discontinuities requires using function spaces that extend the concept of weak derivatives. The gradients of functions of bounded variation are certain measures and the functions may jump across lower-dimensional subsets. The properties of this function space enable the mathematical modeling of fracture and crack formation of materials within the framework of the calculus of variations. Qualitatively, similar model problems arise in image processing to formulate denoising or segmentation of an image. Convergence, error estimation, iterative solution, and implementation of finite element discretizations of these model problems are investigated in this chapter.

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Chapter 11. Elastoplasticity

Abstract

Solid materials react in an elastic way to sufficiently small forces, but when these exceed a threshold, remaining plastic deformations occur. Simple mathematical descriptions lead to nonsmooth evolution problems that can be approximated by sequences of convex minimization problems. Related quasioptimal a priori and a posteriori error estimates for low-order finite element methods are derived. The numerical implementation requires solving a nonlinear, nonsmooth equation at every time step whose realization is based on eliminating the plastic strain. Short codes that realize different types of plastic material behavior are provided.

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Backmatter

Titel: Numerical Methods for Nonlinear Partial Differential Equations
verfasst von: Sören Bartels
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-13797-1
Print ISBN: 978-3-319-13796-4
DOI: https://doi.org/10.1007/978-3-319-13797-1