Numerical experiments involving Galerkin and collocation methods for linear integral equations of the first kind

Journal Article · Tue Mar 01 04:00:00 EST 1983 · J. Comput. Phys.; (United States)

DOI:https://doi.org/10.1016/0021-9991(83)90141-9· OSTI ID:6294295

Allen, Jr, R C; Boland, W R; Wing, G M

Recently effrots have been made to quantify the difficulties inherent in numerically sovling linear Fredholm integral equations of the first kind (J. Integral Equations, to appear.). In particular, the classical quadrature approach, collocation methods, and Galerkin schemes that make use of various orthonormal basis functions have been shown to lead to matrices with high condition numbers. In fact, it has been possible to obtain explicit lower bounds on these condition numbers as a function of the smoothness of the kernel, essentially independent of the choice of orthonormal basis. These bounds all approach infinity as the number of basis functions increases. In this article we present a numerical study of condition numbers arising from collocation and Galerkin methods with step-function and Legendre polynomial bases. The condition number for each kernel and basis set studied is exhibited as a function of the number of basis functions used. The effect that these ill-conditioned matrices have on the accuracy of solutions is demonstrated computationally. The information obtained gives an indication of the efficacy: and the dangers: of the collocation and Galerkin schemes in practical situations.

• Research Organization: Department of Mathematics and Statistics, University of New Mexico, Albuquerque, New Mexico 87131
• OSTI ID: 6294295
• Journal Information: J. Comput. Phys.; (United States), Journal Name: J. Comput. Phys.; (United States) Vol. 49:3; ISSN JCTPA
• Country of Publication: United States
• Language: English

Related Subjects

99 GENERAL AND MISCELLANEOUS
990200* -- Mathematics & Computers
EQUATIONS
ERRORS
FREDHOLM EQUATION
GALERKIN-PETROV METHOD
INTEGRAL EQUATIONS
ITERATIVE METHODS
KERNELS