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Numerical Solution of Elliptic Differential Equations by Reduction to the Interface

  • Book
  • © 2004

Overview

Authors:
  1. Boris N. Khoromskij

    1. Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, Germany

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  2. Gabriel Wittum

    1. Interdisziplinäres Zentrum für Wissenschaftliches Rechnen (IWR), Universität Heidelberg, Heidelberg, Germany

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

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Table of contents (9 chapters)

  1. Front Matter

    Pages I-XI
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  2. Finite Element Method for Elliptic PDEs

    • Boris N. Khoromskij, Gabriel Wittum
    Pages 1-35

  3. Elliptic Poincaré-Steklov Operators

    • Boris N. Khoromskij, Gabriel Wittum
    Pages 37-62

  4. Iterative Substructuring Methods

    • Boris N. Khoromskij, Gabriel Wittum
    Pages 63-81

  5. Multilevel Methods

    • Boris N. Khoromskij, Gabriel Wittum
    Pages 83-95

  6. Robust Preconditioners for Equations with Jumping Anisotropic Coefficients

    • Boris N. Khoromskij, Gabriel Wittum
    Pages 97-124

  7. Frequency Filtering Techniques

    • Boris N. Khoromskij, Gabriel Wittum
    Pages 125-159

  8. Data-sparse Approximation to the Schur Complement for Laplacian

    • Boris N. Khoromskij, Gabriel Wittum
    Pages 161-187

  9. Discrete Poincaré-Steklov Mappings for Biharmonic and Lamé Equations

    • Boris N. Khoromskij, Gabriel Wittum
    Pages 189-208

  10. Interface Reduction for the Stokes Equation

    • Boris N. Khoromskij, Gabriel Wittum
    Pages 209-277

  11. Back Matter

    Pages 279-299
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Keywords

About this book

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.

Authors and Affiliations

  • Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, Germany

    Boris N. Khoromskij

  • Interdisziplinäres Zentrum für Wissenschaftliches Rechnen (IWR), Universität Heidelberg, Heidelberg, Germany

    Gabriel Wittum

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