It has long been known that many elliptic partial differential equations can be reformulated as Fredholm integral equations of the second kind on the boundaries of their domains. The kernels of the resulting integral equations are weakly singular, which has historically made their numerical solution somewhat onerous, requiring the construction of detailed and typically sub-optimal quadrature formulas. Recently, a numerical algorithm for constructing generalized Gaussian quadratures was discovered which, given 2n essentially arbitrary functions, constructs a unique n-point quadrature that integrates them to machine precision, solving the longstanding problem posed by singular kernels.

When the domains have corners, the solutions themselves are also singular. In fact, they are known to be representable, to order n, by a linear combination (expansion) of n known singular functions. In order to solve the integral equation accurately, it is necessary to construct a discretization such that the mapping (in the L^2-sense) from the values at the discretization points to the corresponding n expansion coefficients is well-conditioned. In this talk, we present exactly such an algorithm, which is optimal in the sense that, given n essentially arbitrary functions, it produces n discretization points, and for which the resulting interpolation formulas have condition numbers extremely close to one.

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