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On preconditioners for the Laplace double‐layer in 2D
Author(s) -
Quaife Bryan,
Biros George
Publication year - 2015
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.1940
Subject(s) - mathematics , multigrid method , discretization , solver , integral equation , fast multipole method , laplace transform , matrix (chemical analysis) , domain decomposition methods , convergence (economics) , linear system , condition number , relaxation (psychology) , mathematical analysis , partial differential equation , mathematical optimization , multipole expansion , finite element method , eigenvalues and eigenvectors , physics , materials science , quantum mechanics , economics , composite material , thermodynamics , economic growth , psychology , social psychology
SUMMARY The discretization of the double‐layer potential integral equation for the interior Dirichlet–Laplace problem in a domain with smooth boundary results in a linear system that has a bounded condition number. Thus, the number of iterations required for the convergence of a Krylov method is, asymptotically, independent of the discretization size N . Using the fast multipole method to accelerate the matrix–vector products, we obtain an optimal O ( N ) solver. In practice, however, when the geometry is complicated, the number of Krylov iterations can be quite large—to the extend that necessitates the use of preconditioning. We summarize the different methodologies that have appeared in the literature (single‐grid, multigrid, approximate sparse inverses), and we propose a new class of preconditioners based on a fast multipole method‐based spatial decomposition of the double‐layer operator. We present an experimental study in which we compare the different approaches, and we discuss the merits and shortcomings of our approach. Our method can be easily extended to other second‐kind integral equations with non‐oscillatory kernels in two and three dimensions. Copyright © 2014 John Wiley & Sons, Ltd.