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Adaptive Quadrature by Expansion for Layer Potential Evaluation in Two Dimensions
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.ORCID iD: 0000-0001-7425-8029
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.ORCID iD: 0000-0002-4290-1670
2018 (English)In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 40, no 3, p. A1225-A1249Article in journal (Refereed) Published
Abstract [en]

When solving partial differential equations using boundary integral equation methods, accurate evaluation of singular and nearly singular integrals in layer potentials is crucial. A recent scheme for this is quadrature by expansion (QBX), which solves the problem by locally approximating the potential using a local expansion centered at some distance from the source boundary. In this paper we introduce an extension of the QBX scheme in two dimensions (2D) denoted AQBX—adaptive quadrature by expansion—which combines QBX with an algorithm for automated selection of parameters, based on a target error tolerance. A key component in this algorithm is the ability to accurately estimate the numerical errors in the coefficients of the expansion. Combining previous results for flat panels with a procedure for taking the panel shape into account, we derive such error estimates for arbitrarily shaped boundaries in 2D that are discretized using panel-based Gauss–Legendre quadrature. Applying our scheme to numerical solutions of Dirichlet problems for the Laplace and Helmholtz equations, and also for solving these equations, we find that the scheme is able to satisfy a given target tolerance to within an order of magnitude, making it useful for practical applications. This represents a significant simplification over the original QBX algorithm, in which choosing a good set of parameters can be hard.

Place, publisher, year, edition, pages
Society for Industrial and Applied Mathematics, 2018. Vol. 40, no 3, p. A1225-A1249
Keywords [en]
boundary integral equation, adaptive, quadrature, nearly singular, error estimate
National Category
Computational Mathematics
Research subject
Applied and Computational Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-227312DOI: 10.1137/17M1121615ISI: 000436986000021Scopus ID: 2-s2.0-85044166523OAI: oai:DiVA.org:kth-227312DiVA, id: diva2:1204368
Funder
Göran Gustafsson Foundation for Research in Natural Sciences and MedicineSwedish Research Council, 2015-04998
Note

QC 20180508

Available from: 2018-05-07 Created: 2018-05-07 Last updated: 2018-07-16Bibliographically approved

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af Klinteberg, LudvigTornberg, Anna-Karin
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