One of the standard methods for the solution of elliptic boundary value problems calls for reformulating them as systems of integral equations. The integral operators that arise in this fashion typically have singular kernels, and, in many cases of interest, the solutions of these equations are themselves singular. This makes the accurate discretization of the systems of integral equations arising from elliptic boundary value problems challenging.
Over the last decade, Generalized Gaussian quadrature rules, which are n-point quadrature rules that are exact for a collection of 2n functions, have emerged as one of the most effective tools for discretizing singular integral equations. Among other things, they have been used to accelerate the discretization of singular integral operators on curves, to enable the accurate discretization of singular integral operators on complex surfaces and to greatly reduce the cost of representing the (singular) solutions of integral equations given on planar domains with corners.
We will first briefly outline a standard method for the discretization of integral operators given on curves which is highly amenable to acceleration through generalized Gaussian quadratures. We will then describe a numerical procedure for the construction of Generalized Gaussian quadrature rules.
Much of this is joint work with Zydrunas Gimbutas (NIST Boulder) and Vladimir Rokhlin (Yale University).
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- Computational Mathematics and Applications Seminar