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af Klinteberg, Ludvig; Tornberg, Anna-Karin

Error estimation for quadrature by expansion in layer potential evaluation. (English) Zbl 1367.65154

Adv. Comput. Math. 43, No. 1, 195-234 (2017).
Summary: In boundary integral methods it is often necessary to evaluate layer potentials on or close to the boundary, where the underlying integral is difficult to evaluate numerically. Quadrature by expansion (QBX) is a new method for dealing with such integrals, and it is based on forming a local expansion of the layer potential close to the boundary. In doing so, one introduces a new quadrature error due to nearly singular integration in the evaluation of expansion coefficients. Using a method based on contour integration and calculus of residues, the quadrature error of nearly singular integrals can be accurately estimated. This makes it possible to derive accurate estimates for the quadrature errors related to QBX, when applied to layer potentials in two and three dimensions. As examples we derive estimates for the Laplace and Helmholtz single layer potentials. These results can be used for parameter selection in practical applications.

Cited in 22 Documents

MSC:

65N15 Error bounds for boundary value problems involving PDEs
65N38 Boundary element methods for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
65D32 Numerical quadrature and cubature formulas
41A55 Approximate quadratures

Keywords:

nearly singular; quadrature; layer potential; error estimate; Lapalce equation; Helmholtz equation; numerical example; boundary integral methods; contour integration

Software:

DLMF
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References:

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