Error estimation for quadrature by expansion in layer potential evaluation
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Error estimation for quadrature by expansion in layer potential evaluation Ludvig af Klinteberg1
· Anna-Karin Tornberg1
Received: 20 April 2016 / Accepted: 21 September 2016 © The Author(s) 2016. This article is published with open access at Springerlink.com
Abstract 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. Keywords Nearly singular · Quadrature · Layer potential · Error estimate Mathematics Subject Classification (2010) 65D30 · 65D32 · 65G99
Communicated by: Leslie Greengard Ludvig af Klinteberg
[email protected] 1
Numerical Analysis, Department of Mathematics, KTH Royal Institute of Technology, 100 44 Stockholm, Sweden
L. af Klinteberg and A. -K. Tornberg
1 Introduction At the core of boundary integral equation (BIE) methods for partial differential equations (PDEs) lies the representation of a solution u as a layer potential, u(x) =
G(x, y)σ (y)dSy ,
(1)
where is a smooth, closed contour (in R2 ) or surface (in R3 ), σ (y) is a smooth density defined on , and G(x, y) is a Green’s function associated with the current PDE of interest. The Green’s function is typically singular along the diagonal x = y, which leads to the integrand being singular for x ∈ . We will refer to this as a singular integral. The layer potential u is smooth on either side of , but the error when computing it using a regular quadrature method grows exponentially as x approaches . This is because G develops an increasingly sharp peak, which requires more and more quadrature points to be accurately resolved, even though it is smooth. In this case we say that the integral is nearly singular, since we evaluate G close to its singularity. For an introduction to methods for nearly singular integration, we refer to [11, 14] and the references therein. In this paper we discuss the use of residue calculus for estimating the error committed when computing a nearly singular integral using a regular quadrature method. The error estimates are derived in the limit n → ∞, n being the number of discrete quadrature points, but turn out to be accurate also for moderately large n. Throughout we shall use the symbol to mean “asymptotically equal to
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