## Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and ApplicationsMathematicsisplayinganevermoreimportantroleinthephysicalandbiol- ical sciences, provoking a blurring of boundaries between scienti?c disciplines and a resurgence of interest in the modern as well as the classical techniques of applied mathematics. This renewal of interest, both in research and tea- ing, has led to the establishment of the series Texts in Applied Mathematics (TAM). The development of new courses is a natural consequence of a high level of excitement on the research frontier as newer techniques, such as numerical and symbolic computer systems, dynamical systems, and chaos, mix with and reinforce the traditional methods of applied mathematics. Thus, the purpose ofthistextbookseriesistomeetthecurrentandfutureneedsoftheseadvances and to encourage the teaching of new courses. TAM will publish textbooks suitable for use in advanced undergraduate and beginning graduate courses, and will complement the Applied Mat- matical Sciences (AMS) series, which will focus on advanced textbooks and research-level monographs. Pasadena, California J.E. Marsden Providence, Rhode Island L. Sirovich College Park, Maryland S.S. Antman Preface The algorithms, methods, and Matlab implementations described in this text have been developed during almost a decade of collaboration. During this time we have worked to simplify the basic methods and make the ideas more accessible to a broad audience. Many people, both students and colleagues, have helped during the development of this project and we are grateful for all their suggestions and input. |

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### Contents

ntroduction | 1 |

A brief account of history | 11 |

Scope of text and audience | 17 |

aking it work in one dimension 43 | 42 |

Elementwise operations | 51 |

Getting the grid together and computing the metric | 57 |

Dealing with time | 63 |

Exercises | 72 |

Error estimates for nonlinear problems with smooth solutions | 135 |

Jeyond one dimension | 169 |

ligherorder equations 243 | 242 |

The compressible NavierStokes equations | 314 |

pectral properties of discontinuous Galerkin operators | 331 |

The Maxwell eigenvalue problem | 343 |

Curvilinear elements and nonconforming discretizations | 373 |

nto the third dimension | 407 |

Stability | 83 |

Nonlinear problems | 115 |

Aliasing instabilities and filter stabilization | 123 |

Jacobi polynomials and beyond | 445 |

Software variables and helpful scripts 461 | 460 |

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accuracy applied approach approximation assume basis boundary conditions central flux Chapter choice complex components compute connectivity conservation consider constant construct continuous convergence coordinates defined depends derivatives detail differentiation discrete discussed domain Dose edge eigenvalues element equations error evaluate exact example extension face field filter finite flux formulation function Gauss given global gradient grid illustrate implementation initial integration interpolation introduced limiter linear mapping matrix mesh method modes natural nodal nodes nonlinear normal numerical flux obtained one-dimensional operator penalty points polynomial pressure problem properties recover reference represent scheme shown side similar simple smooth solution solver solving space stability Table timestep tion triangle upwind values variables vector vertices volume wave yields