Mathematics – Numerical Analysis
Scientific paper
2011-12-29
Mathematics
Numerical Analysis
Scientific paper
The note describes the construction of high-order accurate Nystrom discretizations for the Boundary Integral Equations (BIEs) associated with the Laplace and Helmholtz equations in the plane. Only smooth boundaries are considered, but some of the techniques described can be extended to the piece-wise smooth case. The quadrature nodes are equispaced, or those associated with a composite Gaussian rule. In either case, a small number of elements in the coefficient matrix near the diagonal need to be modified to attain high-order accuracy. Three different approaches are considered: (1) Kolm-Rokhlin modification to composite Gaussian quadratures, (2) Kapur-Rokhlin modification to the trapezoidal rule and, (3) Alpert modification to the trapezoidal rule. The three approaches are described in detail, and several numerical experiments illustrating their relative advantages and drawbacks are presented.
Hao Sun
Martinsson Per-Gunnar
Young Patrick
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