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A convergence theorem for a class of Nyström methods for weakly singular integral equations on surfaces in $\mathbb {R}^3$

Authors: Oscar Gonzalez and Jun Li
Journal: Math. Comp. 84 (2015), 675-714
MSC (2010): Primary 65R20, 65N38; Secondary 45B05, 31B20
Published electronically: July 2, 2014
MathSciNet review: 3290960
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Abstract: A convergence theorem is proved for a class of Nyström methods for weakly singular integral equations on surfaces in three dimensions. Fredholm equations of the second kind as arise in connection with linear elliptic boundary value problems for scalar and vector fields are considered. In contrast to methods based on product integration, coordinate transformation and singularity subtraction, the family of Nyström methods studied here is based on a local polynomial correction determined by an auxiliary system of moment equations. The polynomial correction is shown to remove the weak singularity in the integral equation and provide control over the approximation error. Convergence results for the family of methods are established under minimal regularity assumptions consistent with classic potential theory. Rates of convergence are shown to depend on the regularity of the problem, the degree of the polynomial correction, and the order of the quadrature rule employed in the discretization. As a corollary, a simple method based on singularity subtraction which has been employed by many authors is shown to be convergent.

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Additional Information

Oscar Gonzalez
Affiliation: Department of Mathematics, The University of Texas at Austin, Austin, Texas 78712

Jun Li
Affiliation: Graduate Program in Computational and Applied Mathematics, The University of Texas at Austin, Austin, Texas 78712
Address at time of publication: Schlumberger Corporation, Houston, Texas

Received by editor(s): December 29, 2011
Received by editor(s) in revised form: June 3, 2013
Published electronically: July 2, 2014
Additional Notes: This work was supported by the National Science Foundation.
Article copyright: © Copyright 2014 American Mathematical Society