Partial Differential Equations and the Finite Element Method

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John Wiley & Sons, 16 dic 2005 - 512 pagine
A systematic introduction to partial differential

equations and modern finite element methods for their efficient numerical solution

Partial Differential Equations and the Finite Element Method provides a much-needed, clear, and systematic introduction to modern theory of partial differential equations (PDEs) and finite element methods (FEM). Both nodal and hierachic concepts of the FEM are examined. Reflecting the growing complexity and multiscale nature of current engineering and scientific problems, the author emphasizes higher-order finite element methods such as the spectral or hp-FEM.

A solid introduction to the theory of PDEs and FEM contained in Chapters 1-4 serves as the core and foundation of the publication. Chapter 5 is devoted to modern higher-order methods for the numerical solution of ordinary differential equations (ODEs) that arise in the semidiscretization of time-dependent PDEs by the Method of Lines (MOL). Chapter 6 discusses fourth-order PDEs rooted in the bending of elastic beams and plates and approximates their solution by means of higher-order Hermite and Argyris elements. Finally, Chapter 7 introduces the reader to various PDEs governing computational electromagnetics and describes their finite element approximation, including modern higher-order edge elements for Maxwell's equations.

The understanding of many theoretical and practical aspects of both PDEs and FEM requires a solid knowledge of linear algebra and elementary functional analysis, such as functions and linear operators in the Lebesgue, Hilbert, and Sobolev spaces. These topics are discussed with the help of many illustrative examples in Appendix A, which is provided as a service for those readers who need to gain the necessary background or require a refresher tutorial. Appendix B presents several finite element computations rooted in practical engineering problems and demonstrates the benefits of using higher-order FEM.

Numerous finite element algorithms are written out in detail alongside implementation discussions. Exercises, including many that involve programming the FEM, are designed to assist the reader in solving typical problems in engineering and science.

Specifically designed as a coursebook, this student-tested publication is geared to upper-level undergraduates and graduate students in all disciplines of computational engineeringand science. It is also a practical problem-solving reference for researchers, engineers, and physicists.

Dall'interno del libro

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Sommario

1 Partial Differential Equations
1
2 Continuous Elements for 1D Problems
45
3 General Concept of Nodal Elements
103
4 Continuous Elements for 2D Problems
125
5 Transient Problems and ODE Solvers
167
6 Beam and Plate Bending Problems
209
7 Equations of Electromagnetics
269
Appendix A Basics of Functional Analysis
319
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Pagina 1 - We recall that an nxn matrix A is said to be positive definite if vTAv > 0 for all non-zero v € R".
Pagina 340 - A is diagonalizable if and only if it has n linearly independent eigenvectors. In that case, the diagonal matrix D, similar to A is given by /X, 0 0 0 X2 0 0 0 X, D = \ \0 0 0 ••• A.,,/ where X, , X2, ..., Xn are the eigenvalues of A.
Pagina 65 - It is left to the reader as an exercise to verify that the module generated by a,b,c,d with the above relations give A/(2t - l,t — 2) ® A.
Pagina 83 - S consists of three arrays: 1. Array A of length NNZ: This is a real-valued array containing all nonzero entries of the matrix S listed from the left to the right, starting with the first and ending with the last row. 2. Array IA of length N + 1: This is an integer array, IA[l] = 1.

Informazioni sull'autore (2005)

PAVEL SOLÍN, PhD, is Associate Professor in the Department of Mathematical Sciences at The University of Texas at El Paso. Prior to this appointment, Dr. Solin was a postdoctoral research associate at the Johannes Kepler University (Linz, Austria), The University of Texas at Austin, and Rice University (Houston, Texas). He received his PhD from the Charles University in Prague, Czech Republic, in 1999. Dr. Sol?n is a coauthor of the monograph Higher-Order Finite Element Methods.

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