Asymptotic analysis for close evaluation of layer potentials
We study the evaluation of layer potentials close to the domain boundary. Accurate evaluation of layer potentials near boundaries is needed in many applications, including fluid-structure interactions and near-field scattering in nano-optics. When numerically evaluating layer potentials, it is natur...
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Published in | Journal of computational physics Vol. 355; pp. 327 - 341 |
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Main Authors | , , |
Format | Journal Article |
Language | English |
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15.02.2018
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Abstract | We study the evaluation of layer potentials close to the domain boundary. Accurate evaluation of layer potentials near boundaries is needed in many applications, including fluid-structure interactions and near-field scattering in nano-optics. When numerically evaluating layer potentials, it is natural to use the same quadrature rule as the one used in the Nyström method to solve the underlying boundary integral equation. However, this method is problematic for evaluation points close to boundaries. For a fixed number of quadrature points, N, this method incurs O(1) errors in a boundary layer of thickness O(1/N). Using an asymptotic expansion for the kernel of the layer potential, we remove this O(1) error. We demonstrate the effectiveness of this method for interior and exterior problems for Laplace's equation in two dimensions.
•An accurate method for the close evaluation of layer potentials is developed.•Asymptotic expansions are used to capture the behavior of the peaked kernel.•The method combines these expansions with numerical integration to achieve accuracy.•Results are shown for single- and double-layer potentials for 2D Laplace's equation. |
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AbstractList | We study the evaluation of layer potentials close to the domain boundary. Accurate evaluation of layer potentials near boundaries is needed in many applications, including fluid-structure interactions and near-field scattering in nano-optics. When numerically evaluating layer potentials, it is natural to use the same quadrature rule as the one used in the Nyström method to solve the underlying boundary integral equation. However, this method is problematic for evaluation points close to boundaries. For a fixed number of quadrature points, N, this method incurs O(1) errors in a boundary layer of thickness O(1/N). Using an asymptotic expansion for the kernel of the layer potential, we remove this O(1) error. We demonstrate the effectiveness of this method for interior and exterior problems for Laplace's equation in two dimensions. We study the evaluation of layer potentials close to the domain boundary. Accurate evaluation of layer potentials near boundaries is needed in many applications, including fluid-structure interactions and near-field scattering in nano-optics. When numerically evaluating layer potentials, it is natural to use the same quadrature rule as the one used in the Nyström method to solve the underlying boundary integral equation. However, this method is problematic for evaluation points close to boundaries. For a fixed number of quadrature points, N, this method incurs O(1) errors in a boundary layer of thickness O(1/N). Using an asymptotic expansion for the kernel of the layer potential, we remove this O(1) error. We demonstrate the effectiveness of this method for interior and exterior problems for Laplace's equation in two dimensions. •An accurate method for the close evaluation of layer potentials is developed.•Asymptotic expansions are used to capture the behavior of the peaked kernel.•The method combines these expansions with numerical integration to achieve accuracy.•Results are shown for single- and double-layer potentials for 2D Laplace's equation. Accurate evaluation of layer potentials near boundaries is needed in many applications, including fluid-structure interactions and near-field scattering in nano-optics. When numerically evaluating layer potentials, it is natural to use the same quadrature rule as the one used in the Nyström method to solve the underlying boundary integral equation. However, this method is problematic for evaluation points close to boundaries. For a fixed number of quadrature points, N , this method incurs O(1) errors in a boundary layer of thickness O(1/N). Using an asymp-totic expansion for the kernel of the layer potential, we remove this O(1) error. We demonstrate the effectiveness of this method for interior and exterior problems for Laplace's equation in two dimensions. |
Author | Carvalho, Camille Kim, Arnold D. Khatri, Shilpa |
Author_xml | – sequence: 1 givenname: Camille surname: Carvalho fullname: Carvalho, Camille email: ccarvalho3@ucmerced.edu – sequence: 2 givenname: Shilpa surname: Khatri fullname: Khatri, Shilpa email: skhatri3@ucmerced.edu – sequence: 3 givenname: Arnold D. surname: Kim fullname: Kim, Arnold D. email: adkim@ucmerced.edu |
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Cites_doi | 10.1016/j.jcp.2013.06.027 10.1007/BF01061258 10.1038/nphoton.2014.228 10.1016/0016-0032(87)90369-3 10.1021/nn7003734 10.1016/0895-7177(91)90068-I 10.1021/nl802317d 10.1137/15M1043066 10.1137/120900253 10.1016/j.jcp.2010.12.010 10.4208/cicp.030815.240216a 10.1007/BF02091779 10.1137/140990826 10.1016/j.jcp.2007.11.024 10.1038/nphoton.2010.237 10.1137/S0036142999362845 10.1109/22.6086 |
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Keywords | Close evaluations Laplace's equation Layer potentials Nearly singular integrals Boundary integral equations |
Language | English |
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Phys. doi: 10.1016/j.jcp.2007.11.024 contributor: fullname: Helsing – volume: 5 start-page: 83 year: 2011 ident: 10.1016/j.jcp.2017.11.015_br0240 article-title: Antennas for light publication-title: Nat. Photonics doi: 10.1038/nphoton.2010.237 contributor: fullname: Novotny – year: 1992 ident: 10.1016/j.jcp.2017.11.015_br0270 contributor: fullname: Strauss – volume: 38 start-page: 1902 year: 2001 ident: 10.1016/j.jcp.2017.11.015_br0060 article-title: A method for computing nearly singular integrals publication-title: SIAM J. Numer. Anal. doi: 10.1137/S0036142999362845 contributor: fullname: Beale – year: 2017 ident: 10.1016/j.jcp.2017.11.015_br0090 article-title: Local analysis of near fields in acoustic scattering contributor: fullname: Carvalho – volume: 36 start-page: 1386 year: 1988 ident: 10.1016/j.jcp.2017.11.015_br0130 article-title: Strongly convergent Green's function expansions for rectangularly shielded microstrip lines publication-title: IEEE Trans. Microw. Theory Tech. doi: 10.1109/22.6086 contributor: fullname: Fikioris |
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Snippet | We study the evaluation of layer potentials close to the domain boundary. Accurate evaluation of layer potentials near boundaries is needed in many... Accurate evaluation of layer potentials near boundaries is needed in many applications, including fluid-structure interactions and near-field scattering in... |
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SubjectTerms | Analysis of PDEs Asymptotic series Boundary integral equations Boundary layers Close evaluations Computational physics Differential equations Fluid-structure interaction Harmonic analysis Integral equations Integrals Laplace's equation Layer potentials Mathematical Physics Mathematics Nano-optics Nearly singular integrals Numerical Analysis Thickness |
Title | Asymptotic analysis for close evaluation of layer potentials |
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