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Take a Fredholm integral equation $$ u(x) - \lambda \int_{-1}^{1} K(x,y)u(y) \, \mathrm{d} y = f(x) $$ and discretize it via (say) Gaussian quadrature with nodes $\{x_j\}$ and weights $\{w_j\}$ to get $$ u(x) - \lambda \sum_{j} w_j K(x,x_j)u(x_j) = f(x) $$ It feels natural to use the same set of quadrature nodes over $f$ to get a linear system, so $$ u(x_k) - \lambda \sum_{j} w_j K(x_k,x_j)u(x_j) = f(x_k) $$ But if $K$ is singular, we have a big problem on out hands.

What are the normal methods of dealing with this?

Obviously we could use Gaussian quadrature nodes over $y$ and Chebyshev nodes over $x$, but that feels a little awkward; are there better ways?

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    $\begingroup$ Look into the locally corrected Nystrom scheme, which is designed specifically for this problem. $\endgroup$ – sssssssssssss Mar 19 at 11:17
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    $\begingroup$ There is a substantial literature on quadrature in the context of the Boundary Element Method. You might want to look there. $\endgroup$ – Wolfgang Bangerth Mar 19 at 14:59
  • $\begingroup$ To answer this question, we need to see your equation, not the Fredholm equation in general. $\endgroup$ – Alex Trounev Mar 19 at 18:13
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In general, there are three major method families that are used (excluding locally corrected Nyström, as I am not an expert on this one):

  1. Treat the singularity. In this Q/A, I describe a canonical example in electromagnetics community with an EFIE integral, where for a singular integral, the singular part is treated analytically, while the rest is computed via regular quadratures. That Q/A includes a link to a paper treating common integrals for integral-equation methods, and there are some follow-ups.
  2. Use special quadrature rules that are tailored to capture singularities. The paper by J. Ma, V. Rokhlin, and S. Wandzura, "Generalized Gaussian quadrature rules for systems of arbitrary functions," SIAM J. Num. Analysis, vol. 33, no. 3, pp. 971–996, 1996 is one of the papers I often use, and it can serve as a great entry point in this topic.

Generally, using mixed quadratures can provide somewhat decent results in some cases within heavy adaptive integration (that's the third family); however, from my experience (and there might be references found) it is unable to provide stable high-accurate results or even compete over the aforementioned methods in terms of computational speed.

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