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Circulant preconditioners for ill‐conditioned boundary integral equations from potential equations
Author(s) -
Chan Raymond H.,
Sun HaiWei,
Ng WingFai
Publication year - 1998
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/(sici)1097-0207(19981230)43:8<1505::aid-nme483>3.0.co;2-q
Subject(s) - circulant matrix , mathematics , integral equation , conjugate gradient method , discretization , mathematical analysis , matrix (chemical analysis) , convergence (economics) , rate of convergence , condition number , convolution (computer science) , boundary value problem , eigenvalues and eigenvectors , mathematical optimization , algorithm , computer science , physics , key (lock) , materials science , computer security , quantum mechanics , machine learning , artificial neural network , economics , composite material , economic growth
In this paper, we consider solving potential equations by the boundary integral equation approach. The equations so derived are Fredholm integral equations of the first kind and are known to be ill‐conditioned. Their discretized matrices are dense and have condition numbers growing like O ( n ) where n is the matrix size. We propose to solve the equations by the preconditioned conjugate gradient method with circulant integral operators as preconditioners. These are convolution operators with periodic kernels and hence can be inverted efficiently by using fast Fourier transforms. We prove that the preconditioned systems are well conditioned, and hence the convergence rate of the method is linear. Numerical results for two types of regions are given to illustrate the fast convergence. © 1998 John Wiley & Sons, Ltd.