## Frontiers of Numerical Analysis: Durham 2004This book contains detailed lecture notes on four topics at the forefront of current research in computational mathematics. Each set of notes presents a self-contained guide to a current research area and has an extensive bibliography. In addition, most of the notes contain detailed proofs of the key results. The notes start from a level suitable for first year graduate students in applied mathematics, mathematical analysis or numerical analysis, and proceed to current research topics. The reader should therefore be able to gain quickly an insight into the important results and techniques in each area without recourse to the large research literature. Current (unsolved) problems are also described and directions for future research are given. This book is also suitable for professional mathematicians who require a succint and accurate account of recent research in areas parallel to their own, and graduates in mathematical sciences. |

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### Contents

1 | |

Wavelets | 13 |

Problems in Wavelet Coordinates | 33 |

Iterative Solution | 44 |

References | 60 |

Polynomial Reproducing Systems | 69 |

Partition of Unity Method and Generalized FEM | 93 |

A Results from Analysis | 122 |

Theory and Applications of Smoothed Particle Hydrodynamics143 | 142 |

Lagrangian SPH | 167 |

SPH Heat Conduction | 177 |

Viscosity | 183 |

References | 193 |

Efficient Implementation | 212 |

Parallelization | 243 |

References | 258 |

### Other editions - View all

Frontiers in Numerical Analysis: Durham 2002 James Blowey,Alan Craig,Tony Shardlow Limited preview - 2012 |

Frontiers in Numerical Analysis: Durham 2002 James Blowey,Alan Craig,Tony Shardlow Limited preview - 2003 |

### Common terms and phrases

approximation properties approximation spaces Assumption aWab basis functions biorthogonal bound boundary value problems classical FEM coefficients complexity computational condition numbers constant control problems convergence cover construction cover patches decomposition deﬁned denotes density derived differential Dirichlet boundary discretization elliptic error estimate example Exercise Figure finite element method formulation function spaces Galerkin given global grid Hence integration cells interpolation iteration kernel Lagrangian Lemma linear system Lipschitz domain Math mesh meshfree methods minimal moving least squares neighbours nodes Note operator optimal overlap parallel parameter particle partition of unity piecewise point set polynomial degree processor Proof quadrature rule respect result saddle point saddle point problems satisﬁes satisfy scheme Section shape functions singularity functions smooth smoother Sobolev spaces solution solve sparse grid step stiffness matrix subspace Theorem unity functions wavelet weight functions