Close evaluation of layer potentials in three dimensions
International audience We present a simple and effective method for evaluating double-and single-layer potentials for Laplace's equation in three dimensions close to the boundary. The close evaluation of these layer potentials is challenging because they are nearly singular integrals. The metho...
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ftccsdartic:oai:HAL:hal-02970692v1 2023-05-15T17:39:58+02:00 Close evaluation of layer potentials in three dimensions Khatri, Shilpa Kim, Arnold, Cortez, R Carvalho, Camille University of California Merced (UC Merced) University of California (UC) 2020 https://hal.archives-ouvertes.fr/hal-02970692 https://hal.archives-ouvertes.fr/hal-02970692/document https://hal.archives-ouvertes.fr/hal-02970692/file/KKCC3Drevision.pdf en eng HAL CCSD hal-02970692 https://hal.archives-ouvertes.fr/hal-02970692 https://hal.archives-ouvertes.fr/hal-02970692/document https://hal.archives-ouvertes.fr/hal-02970692/file/KKCC3Drevision.pdf info:eu-repo/semantics/OpenAccess ISSN: 0021-9991 Journal of Computational Physics https://hal.archives-ouvertes.fr/hal-02970692 Journal of Computational Physics, 2020 Nearly singular integrals close evaluation problem potential theory boundary inte- gral equations numerical quadrature [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA] [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph] info:eu-repo/semantics/article Journal articles 2020 ftccsdartic 2022-12-04T00:49:09Z International audience We present a simple and effective method for evaluating double-and single-layer potentials for Laplace's equation in three dimensions close to the boundary. The close evaluation of these layer potentials is challenging because they are nearly singular integrals. The method we propose is based on writing these layer potentials in spherical coordinates where the point at which their kernels are peaked maps to the north pole. An N-point Gauss-Legendre quadrature rule is used for integration with respect to the the polar angle rather than the cosine of the polar angle. A 2N-point periodic trapezoid rule is used to compute the integral with respect to the azimuthal angle which acts as a natural and effective averaging operation in this coordinate system. The numerical method resulting from combining these two quadrature rules in this rotated coordinate system yields results that are consistent with asymptotic behaviors of the double-and single-layer potentials at close evaluation distances. In particular, we show that the error in computing the double-layer potential, after applying a subtraction method, is quadratic with respect to the evaluation distance from the boundary, and the error is linear for the single-layer potential. We improve upon the single-layer potential by introducing an alternate approximation based on a perturbation expansion and obtain an error that is quadratic with respect to the evaluation distance from the boundary. Article in Journal/Newspaper North Pole Archive ouverte HAL (Hyper Article en Ligne, CCSD - Centre pour la Communication Scientifique Directe) North Pole |
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Open Polar |
collection |
Archive ouverte HAL (Hyper Article en Ligne, CCSD - Centre pour la Communication Scientifique Directe) |
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ftccsdartic |
language |
English |
topic |
Nearly singular integrals close evaluation problem potential theory boundary inte- gral equations numerical quadrature [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA] [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph] |
spellingShingle |
Nearly singular integrals close evaluation problem potential theory boundary inte- gral equations numerical quadrature [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA] [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph] Khatri, Shilpa Kim, Arnold, Cortez, R Carvalho, Camille Close evaluation of layer potentials in three dimensions |
topic_facet |
Nearly singular integrals close evaluation problem potential theory boundary inte- gral equations numerical quadrature [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA] [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph] |
description |
International audience We present a simple and effective method for evaluating double-and single-layer potentials for Laplace's equation in three dimensions close to the boundary. The close evaluation of these layer potentials is challenging because they are nearly singular integrals. The method we propose is based on writing these layer potentials in spherical coordinates where the point at which their kernels are peaked maps to the north pole. An N-point Gauss-Legendre quadrature rule is used for integration with respect to the the polar angle rather than the cosine of the polar angle. A 2N-point periodic trapezoid rule is used to compute the integral with respect to the azimuthal angle which acts as a natural and effective averaging operation in this coordinate system. The numerical method resulting from combining these two quadrature rules in this rotated coordinate system yields results that are consistent with asymptotic behaviors of the double-and single-layer potentials at close evaluation distances. In particular, we show that the error in computing the double-layer potential, after applying a subtraction method, is quadratic with respect to the evaluation distance from the boundary, and the error is linear for the single-layer potential. We improve upon the single-layer potential by introducing an alternate approximation based on a perturbation expansion and obtain an error that is quadratic with respect to the evaluation distance from the boundary. |
author2 |
University of California Merced (UC Merced) University of California (UC) |
format |
Article in Journal/Newspaper |
author |
Khatri, Shilpa Kim, Arnold, Cortez, R Carvalho, Camille |
author_facet |
Khatri, Shilpa Kim, Arnold, Cortez, R Carvalho, Camille |
author_sort |
Khatri, Shilpa |
title |
Close evaluation of layer potentials in three dimensions |
title_short |
Close evaluation of layer potentials in three dimensions |
title_full |
Close evaluation of layer potentials in three dimensions |
title_fullStr |
Close evaluation of layer potentials in three dimensions |
title_full_unstemmed |
Close evaluation of layer potentials in three dimensions |
title_sort |
close evaluation of layer potentials in three dimensions |
publisher |
HAL CCSD |
publishDate |
2020 |
url |
https://hal.archives-ouvertes.fr/hal-02970692 https://hal.archives-ouvertes.fr/hal-02970692/document https://hal.archives-ouvertes.fr/hal-02970692/file/KKCC3Drevision.pdf |
geographic |
North Pole |
geographic_facet |
North Pole |
genre |
North Pole |
genre_facet |
North Pole |
op_source |
ISSN: 0021-9991 Journal of Computational Physics https://hal.archives-ouvertes.fr/hal-02970692 Journal of Computational Physics, 2020 |
op_relation |
hal-02970692 https://hal.archives-ouvertes.fr/hal-02970692 https://hal.archives-ouvertes.fr/hal-02970692/document https://hal.archives-ouvertes.fr/hal-02970692/file/KKCC3Drevision.pdf |
op_rights |
info:eu-repo/semantics/OpenAccess |
_version_ |
1766140726504062976 |