## Partial Differential Equations and the Finite Element MethodA 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. |

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### Contents

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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afﬁne Algorithm analogously Banach space basis functions bilinear bubble functions coefﬁcients connectivity arrays Consider constant continuous convergence deﬁned Deﬁnition degrees of freedom Dirichlet boundary conditions discrete edge elements Elem elliptic equivalent Euler method exact solution example Exercise existence and uniqueness Fekete points ﬁnd ﬁnite element mesh ﬁrst Gaussian quadrature Hermite elements hierarchic shape functions higher-order Hilbert space inequality inner product inner product space integral interface Lagrange nodal Lemma linear algebraic linear forms linear operator linear space Lobatto lowest-order maximum norm Maxwell’s equations nodal basis nodal interpolant nodal points nodal shape functions normed space obtain orthogonal Paragraph PDEs piecewise-afﬁne polynomial degree polynomial space Proof Q C Rd reference domain reference map right-hand side RK methods satisﬁes second-order sequence Sobolev space space V I stiffness matrix subspace sufﬁciently Theorem triangular unisolvent V-elliptic vector vertex basis functions vertex functions weak formulation zero

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Page 1 - We recall that an nxn matrix A is said to be positive definite if vTAv > 0 for all non-zero v € R".