## Computing with hp-ADAPTIVE FINITE ELEMENTS: Volume 1 One and Two Dimensional Elliptic and Maxwell ProblemsOffering the only existing finite element (FE) codes for Maxwell equations that support hp refinements on irregular meshes, Computing with hp-ADAPTIVE FINITE ELEMENTS: Volume 1. One- and Two-Dimensional Elliptic and Maxwell Problems presents 1D and 2D codes and automatic hp adaptivity. This self-contained source discusses the theory and implementation of hp-adaptive FE methods, focusing on projection-based interpolation and the corresponding hp-adaptive strategy. The book is split into three parts, progressing from simple to more advanced problems. Part I examines the hp elements for the standard 1D model elliptic problem. The author develops the variational formulation and explains the construction of FE basis functions. The book then introduces the 1D code (1Dhp) and automatic hp adaptivity. This first part ends with a study of a 1D wave propagation problem. In Part II, the book proceeds to 2D elliptic problems, discussing two model problems that are slightly beyond standard-level examples: 3D axisymmetric antenna problem for Maxwell equations (example of a complex-valued, indefinite problem) and 2D elasticity (example of an elliptic system). The author concludes with a presentation on infinite elements - one of the possible tools to solve exterior boundary-value problems. Part III focuses on 2D time-harmonic Maxwell equations. The book explains the construction of the hp edge elements and the fundamental de Rham diagram for the whole family of hp discretizations. Next, it explores the differences between the elliptic and Maxwell versions of the 2D code, including automatic hp adaptivity. Finally, the book presents 2D exterior (radiation and scattering) problems and sample solutions using coupled hp finite/infinite elements. In Computing with hp-ADAPTIVE FINITE ELEMENTS, the information provided, including many unpublished details, aids in solving elliptic and Maxwell problems. |

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### Contents

Chapter 1 1D Model Elliptic Problem | 3 |

Chapter 2 Galerkin Method | 33 |

Chapter 3 1D hp Finite Element Method | 41 |

Chapter 4 1D hp Code | 57 |

Chapter 5 Mesh Refinements in 1D | 71 |

Chapter 6 Automatic hp Adaptivity in 1D | 95 |

Chapter 7 Wave Propagation Problems | 117 |

Part II 2D Elliptic Problems | 133 |

Chapter 15 Examples of Applications | 241 |

Chapter 16 Exterior BoundaryValue Problems | 259 |

Part III 2D Maxwell Problems | 283 |

Chapter 17 2D Maxwell Equations | 285 |

Chapter 18 Edge Elements and the de Rham Diagram | 301 |

Chapter 19 2D Maxwell Code | 323 |

Chapter 20 hp Adaptivity for Maxwell Equations | 337 |

Chapter 21 Exterior Maxwell BoundaryValue Problems | 351 |

Chapter 8 2D Elliptic BoundaryValue Problem | 135 |

Chapter 9 Sobolev Spaces | 149 |

Chapter 10 2D hp Finite Element Mehtod on Regular Meshes | 163 |

Chapter 11 2D hp Code | 185 |

Chapter 12 Geometric Modeling and Mesh Generation | 195 |

Chapter 13 The hp Finite Element Method on hRefined Meshes | 211 |

Chapter 14 Automatic hp Adaptivity in 2D | 227 |

Chapter 22 A Quick Summary and Outlook | 369 |

Appendix A Construction of makefile | 371 |

375 | |

389 | |

Back cover | 399 |

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Computing with hp-ADAPTIVE FINITE ELEMENTS: Volume 1 One and Two Dimensional ... Leszek Demkowicz No preview available - 2006 |

### Common terms and phrases

adaptivity approximation error basis functions bilinear form boundary conditions Cauchy coarse grid coefficients component Comput constant constrained nodes convergence history coordinates corresponding data structure arrays defined Demkowicz denotes determine Dirichlet BC discrete discussed domain element d.o.f. element edges element matrices element shape functions element sons end of loop error estimate exact solution Exercise Figure finite element method flag frontal solver Galerkin method global gradient grid solution h refinements H(curl H(curl)-conforming Helmholtz equation hp-adaptive infinite integration interface interpolation error Jacobi polynomials linear load vector master element Maxwell equations mdle Mech Meth middle node midedge modified element Neumann norm nrdof operator order of approximation parametrization plane wave polynomials of order projection-based interpolation quad rectangle Section seminorm Sobolev spaces solve stiffness matrix test functions theorem tion triangle vanish variational formulation vertex nodes zero