Numerical Solution of Elliptic Differential Equations by Reduction to the Interface

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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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.

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

• 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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