Numerical Solution of Elliptic Differential Equations by Reduction to the Interface / Edition 1

by Boris N. Khoromskij, Gabriel Wittum

Overview

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 modern 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.

Product Details

ISBN-13:	9783540204060
Publisher:	Springer Berlin Heidelberg
Publication date:	03/31/2004
Series:	Lecture Notes in Computational Science and Engineering , #36
Edition description:	Softcover reprint of the original 1st ed. 2004
Pages:	293
Product dimensions:	6.10(w) x 9.25(h) x 0.03(d)

Table of Contents

1. Finite Element Method for Elliptic PDEs.- 2. Elliptic Poincaré-Steklov Operators.- 3. Iterative Substructuring Methods.- 4. Multilevel Methods.- 5. Robust Preconditioners for Equations with Jumping Anisotropic Coefficients.- 6. Frequency Filtering Techniques.- 7. Data-sparse Approximation to the Schur Complement for Laplacian.- 8. Discrete Poincaré-Steklov Mappings for Biharmonic and Lamé Equations.- 9. Interface Reduction for the Stokes Equation.- References.