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Numerical Solution of Elliptic Differential Equations by Reduction to the Interface

  • Boris N. Khoromskij
  • Gabriel Wittum

Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 36)

Table of contents

  1. Front Matter
    Pages I-XI
  2. Boris N. Khoromskij, Gabriel Wittum
    Pages 1-35
  3. Boris N. Khoromskij, Gabriel Wittum
    Pages 37-62
  4. Boris N. Khoromskij, Gabriel Wittum
    Pages 63-81
  5. Boris N. Khoromskij, Gabriel Wittum
    Pages 83-95
  6. Boris N. Khoromskij, Gabriel Wittum
    Pages 97-124
  7. Boris N. Khoromskij, Gabriel Wittum
    Pages 125-159
  8. Boris N. Khoromskij, Gabriel Wittum
    Pages 161-187
  9. Boris N. Khoromskij, Gabriel Wittum
    Pages 189-208
  10. Boris N. Khoromskij, Gabriel Wittum
    Pages 209-277
  11. Back Matter
    Pages 279-299

About this book

Introduction

During the last decade essential progress has been achieved in the analysis and implementation of multilevel/rnultigrid and domain decomposition methods to explore a variety of real world applications. An important trend in mod­ ern numerical simulations is the quick improvement of computer technology that leads to the well known paradigm (see, e. g. , [78,179]): high-performance computers make it indispensable to use numerical methods of almost linear complexity in the problem size N, to maintain an adequate scaling between the computing time and improved computer facilities as N increases. In the h-version of the finite element method (FEM), the multigrid iteration real­ izes an O(N) solver for elliptic differential equations in a domain n c IRd d with N = O(h- ) , where h is the mesh parameter. In the boundary ele­ ment method (BEM) , the traditional panel clustering, fast multi-pole and wavelet based methods as well as the modern hierarchical matrix techniques are known to provide the data-sparse approximations to the arising fully populated stiffness matrices with almost linear cost O(Nr log?Nr), where 1 d Nr = O(h - ) is the number of degrees of freedom associated with the boundary. The aim of this book is to introduce a wider audience to the use of a new class of efficient numerical methods of almost linear complexity for solving elliptic partial differential equations (PDEs) based on their reduction to the interface.

Keywords

Operator Poincaré-Steklov operators data-sparse approximation differential equation elliptic equations finite element method finite element methods multilevel methods partial differential equation

Authors and affiliations

  • Boris N. Khoromskij
    • 1
  • Gabriel Wittum
    • 2
  1. 1.Max-Planck-Institut für Mathematik in den NaturwissenschaftenLeipzigGermany
  2. 2.Interdisziplinäres Zentrum für Wissenschaftliches Rechnen (IWR)Universität HeidelbergHeidelbergGermany

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-642-18777-3
  • Copyright Information Springer-Verlag Berlin Heidelberg 2004
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-540-20406-0
  • Online ISBN 978-3-642-18777-3
  • Series Print ISSN 1439-7358
  • Buy this book on publisher's site
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