Galerkin methods for second kind integral equations with singularities
Author:
Ivan G. Graham
Journal:
Math. Comp. 39 (1982), 519533
MSC:
Primary 65R20; Secondary 45E05, 45L10
MathSciNet review:
669644
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Abstract: This paper discusses the numerical solution of Fredholm integral equations of the second kind which have weakly singular kernels and inhomogeneous terms. Global convergence estimates are derived for the Galerkin and iterated Galerkin methods using splines on arbitrary quasiuniform meshes as approximating subspaces. It is observed that, due to the singularities present in the solution being approximated, the resulting convergence may be slow. It is then shown that convergence will be improved greatly by the use of splines based on a mesh which has been suitably graded to accommodate these singularities. In fact, it is shown that, under suitable conditions, the Galerkin method converges optimally and the iterated Galerkin method is superconvergent. Numerical llustrations are given.
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 F. Chatelin & R. Lebbar, "Superconvergence results for the iterated projection method applied to a second kind Fredholm integral equation and eigenvalue problem." (Preprint.)
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 G. Vainikko & P. Uba, "A piecewise polynomial approximation to the solution of an integral equation with weakly singular kernel," J. Austral. Math. Soc. Ser. B, v. 22, 1981, pp. 431438. MR 626934 (82h:65100)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198206696443
PII:
S 00255718(1982)06696443
Keywords:
Second kind Fredholm integral equation,
weak singularities,
Galerkin method,
iterated Galerkin method,
spline approximation,
graded mesh
Article copyright:
© Copyright 1982
American Mathematical Society
