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Über dieses Buch

This book presents practical applications of the finite element method to general differential equations. The underlying strategy of deriving the finite element solution is introduced using linear ordinary differential equations, thus allowing the basic concepts of the finite element solution to be introduced without being obscured by the additional mathematical detail required when applying this technique to partial differential equations. The author generalizes the presented approach to partial differential equations which include nonlinearities. The book also includes variations of the finite element method such as different classes of meshes and basic functions. Practical application of the theory is emphasised, with development of all concepts leading ultimately to a description of their computational implementation illustrated using Matlab functions. The target audience primarily comprises applied researchers and practitioners in engineering, but the book may also be beneficial for graduate students.

Inhaltsverzeichnis

Frontmatter

Chapter 1. An Overview of the Finite Element Method

Differential equations play a central role in quantitative sciences and are used to describe phenomena as diverse as the flow of traffic through a city, the evolution of an oilfield over many years, the flow of ions through an electrochemical solution and the effect of a drug on a given patient. Most differential equations that are encountered in practice do not possess an analytic solution, that is a solution that can be written in terms of known functions. Under these circumstances, a numerical solution to the differential equation is often sought. The finite element method is one common technique for calculating a numerical solution of a differential equation.

Jonathan Whiteley

Chapter 2. A First Example

In Chap. 1, we explained the fundamental philosophy that underpins the finite element method—that is, the region on which a differential equation is defined is partitioned into smaller regions known as elements, and the solution on each of these elements is approximated using a low-order polynomial function. In this chapter, with the aid of a simple example, we illustrate how this may be done. This overview will require the definition of some terms that the reader may not be familiar with, as well as a few technical details. We will, however, undertake to keep these definitions to a minimum and will focus on the underlying strategy of applying the finite element method without getting bogged down by these technical details. As a consequence, we will inevitably skate over some mathematical rigour, but will make a note to return to these points in later chapters.

Jonathan Whiteley

Chapter 3. Linear Boundary Value Problems

In the previous chapter, we demonstrated how to calculate the finite element solution of a very simple differential equation. The intention of this simple example was to give the reader an overview of the steps required when applying the finite element method. As a consequence, for the sake of clarity, some mathematical rigour was neglected. In this chapter, we introduce more rigour into our description of the finite element method and demonstrate how to calculate the finite element solution of general linear, second-order, boundary value problems.

Jonathan Whiteley

Chapter 4. Higher Order Basis Functions

We claimed, in Chap. 1, that the finite element method is a very flexible technique for calculating the numerical solution of differential equations. One justification for this claim is that the finite element method allows the solution of a differential equation to be approximated, on each element of the computational mesh, using a polynomial of any degree chosen by the user. The material presented in the earlier chapters of this book has focused entirely on finite element solutions that are linear approximations on each element. In this chapter, we extend this material to allow a general polynomial approximation on each element.

Jonathan Whiteley

Chapter 5. Nonlinear Boundary Value Problems

The differential equations investigated in earlier chapters have all been linear differential equations. The finite element discretisation of these equations yielded systems of linear algebraic equations that may be solved using established, robust and reliable linear algebra techniques. In this chapter, we describe the application of the finite element method to nonlinear boundary value problems.

Jonathan Whiteley

Chapter 6. Systems of Ordinary Differential Equations

So far we have only considered scalar boundary value problems, that is, a single differential equation that contains one unknown function u(x). We now generalise the finite element method so that it may be applied to coupled systems of differential equations. We illustrate the basic principles using a simple example, before explaining how these ideas may be generalised for application to more complex systems.

Jonathan Whiteley

Chapter 7. Linear Elliptic Partial Differential Equations

In earlier chapters, we described how to apply the finite element method to ordinary differential equations. For the remainder of this book, we will focus on extending this technique for application to partial differential equations. As with ordinary differential equations, we begin with a simple example to illustrate the key features. We then discuss more complex differential equations in later chapters.

Jonathan Whiteley

Chapter 8. More General Elliptic Problems

In the previous chapter, we explained how to calculate the finite element solution of a very simple elliptic partial differential equation, where Dirichlet boundary conditions were applied on the whole boundary. We develop these ideas in this chapter to allow the application of the finite element method to more general linear, elliptic partial differential equations, and more general boundary conditions.

Jonathan Whiteley

Chapter 9. Quadrilateral Elements

In Chaps. 7 and 8, we calculated the finite element solution of partial differential equations, having partitioned the domain on which the differential equation is defined into a mesh of triangular elements. Triangular elements are not, however, the only shape of elements that can be used when partitioning the domain. In this chapter, we explain how a mesh comprising quadrilateral elements may be used when calculating the finite element solution of a partial differential equation.

Jonathan Whiteley

Chapter 10. Higher Order Basis Functions

In Chap. 4, we described the use of quadratic, and higher order, basis functions when calculating the finite element solution of ordinary differential equations. In this chapter, we will explain how higher order basis functions may be used to calculate the finite element solution of a partial differential equation. We give a concrete example of the use of quadratic basis functions on a mesh of triangular elements. The exercises at the end of the chapter contain examples of higher order basis functions on meshes of both triangular and quadrilateral elements.

Jonathan Whiteley

Chapter 11. Nonlinear Elliptic Partial Differential Equations

In Chap. 5, we explained howPartial differential equationsellipticPartial differential equationsnonlinear to apply the finite element method to nonlinear ordinary differential equations. We saw that calculating the finite element solution of nonlinear differential equations required us to solve a nonlinear system of algebraic equations and discussed how these algebraic equations could be solved. In this chapter, we explain how to apply the finite element method to nonlinear partial differential equations by combining: (i) the material on calculating the finite element solution of linear partial differential equations given in Chaps. 7–10 and (ii) the material on calculating the finite element solution of nonlinear ordinary differential equations in Chap. 5.

Jonathan Whiteley

Chapter 12. Systems of Elliptic Equations

We saw, in Chap. 6, how the finitePartial differential equationsellipticSystems of differential equations element method may be applied to systems of ordinary differential equations. Then, in Chaps. 7–11, we saw how to apply the finite element method to elliptic partial differential equations, using a variety of meshes and basis functions. In this chapter, we combine this material, allowing us to apply the finite element method to systems of elliptic partial differential equations.

Jonathan Whiteley

Chapter 13. Parabolic Partial Differential Equations

We now describe how to apply the finite element to parabolic partial differential equations. This is done by approximating the parabolic partial differential equation by either a sequence of ordinary differential equations or a sequence of elliptic partial differential equations. We may then solve these ordinary differential equations or elliptic partial differential equations using the techniques developed earlier in this book.

Jonathan Whiteley

Backmatter

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