Galerkin methods for second kind integral equations with singularities

Author:
Ivan G. Graham

Journal:
Math. Comp. **39** (1982), 519-533

MSC:
Primary 65R20; Secondary 45E05, 45L10

DOI:
https://doi.org/10.1090/S0025-5718-1982-0669644-3

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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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1982-0669644-3

Keywords:
Second kind Fredholm integral equation,
weak singularities,
Galerkin method,
iterated Galerkin method,
spline approximation,
graded mesh

Article copyright:
© Copyright 1982
American Mathematical Society