Improved Quadrature Formula for a Single-Layer Potential
An improved quadrature formula is derived for a single-layer potential with a smooth density given on a closed or open surface. The formula ensures a uniform approximation of the potential near the surface and preserves the continuity of the potential as the observation point tends to the surface fr...
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| Published in | Computational mathematics and mathematical physics Vol. 63; no. 2; pp. 187 - 201 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Moscow
Pleiades Publishing
01.02.2023
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
| ISSN | 0965-5425 1555-6662 |
| DOI | 10.1134/S0965542523020100 |
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| Abstract | An improved quadrature formula is derived for a single-layer potential with a smooth density given on a closed or open surface. The formula ensures a uniform approximation of the potential near the surface and preserves the continuity of the potential as the observation point tends to the surface from inside the domain. These properties are confirmed by numerical tests. For the potential computed near the surface, the present quadrature formula yields higher accuracy than previously known quadrature rules, which is also confirmed by numerical tests. Additionally, a quadrature formula for the direct value of the single-layer potential on the surface is derived. Numerical tests conducted with this formula confirm its efficiency and accuracy. |
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| AbstractList | An improved quadrature formula is derived for a single-layer potential with a smooth density given on a closed or open surface. The formula ensures a uniform approximation of the potential near the surface and preserves the continuity of the potential as the observation point tends to the surface from inside the domain. These properties are confirmed by numerical tests. For the potential computed near the surface, the present quadrature formula yields higher accuracy than previously known quadrature rules, which is also confirmed by numerical tests. Additionally, a quadrature formula for the direct value of the single-layer potential on the surface is derived. Numerical tests conducted with this formula confirm its efficiency and accuracy. |
| Author | Reznichenko, I. O. Krutitskii, P. A. |
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| Cites_doi | 10.1134/S0012266119090106 10.32917/hmj/1206126877 10.1080/00036819608840484 10.1134/S0012266116090135 10.1007/978-3-642-48860-3 10.4171/ZAA/827 10.1093/imamat/60.3.285 10.1007/s10665-006-9058-x 10.4171/ZAA/788 10.1090/S0033-569X-08-01117-4 10.1155/S0161171298000301 |
| ContentType | Journal Article |
| Copyright | Pleiades Publishing, Ltd. 2023. ISSN 0965-5425, Computational Mathematics and Mathematical Physics, 2023, Vol. 63, No. 2, pp. 187–201. © Pleiades Publishing, Ltd., 2023. Russian Text © The Author(s), 2023, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2023, Vol. 63, No. 2, pp. 230–244. Pleiades Publishing, Ltd. 2023. |
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| DOI | 10.1134/S0965542523020100 |
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| References | PrudnikovA. P.BrychkovYu. A.MarichevO. I.Integrals and Series1986New YorkGordon and Breach0606.33001 KrutitskiiP. A.The 2-dimensional Dirichlet problem in an external domain with cutsZ. Anal. Anwend.199817361378163255110.4171/ZAA/8270907.35036 KrutitskiiP. A.The Neumann problem on wave propagation in a 2-D external domain with cutsJ. Math. Kyoto Univ.19983843945216612040948.35031 KrutitskiiP. A.The Neumann problem for the 2-D Helmholtz equation in a domain, bounded by closed and open curvesInt. J. Math. Math. Sci.199821209216160970810.1155/S01611712980003010910.35040 V. A. Il’in and E. G. Poznyak, Fundamentals of Calculus (Fizmatlit, Moscow, 1973), Part 2 [in Russian]. A. N. Tikhonov and A. A. Samarskii, Equations of Mathematical Physics (Nauka, Moscow, 1999; Dover, New York, 2011). ColtonD.KressR.Integral Equation Methods in Scattering Theory1984New YorkWiley0522.35001 KrutitskiiP. A.FedotovA. D.KolybasovaV. V.Quadrature formula for the simple layer potentialDiffer. Equations20195512261241402804410.1134/S00122661190901061455.65040 ButuzovV. F.KrutitskayaN. Ch.MedvedevG. N.ShishkinA. A.Calculus in Questions and Problems2000MoscowFizmatlit1029.00002 BrebbiaC. A.TellesJ. C. F.WrobelL. C.Boundary Element Techniques: Theory and Applications in Engineering1984BerlinSpringer-Verlag10.1007/978-3-642-48860-30556.73086 P. A. Krutitskii, A. D. Fedotov, and V. V. Kolybasova, Preprint No. 112, IPM RAN (Keldysh Inst. of Applied Mathematics, Russian Academy of Sciences, Moscow, 2019). KrutitskiiP. A.The Dirichlet problem for the dissipative Helmholtz equation in a plane domain bounded by closed and open curvesHiroshima Math. J.199828149168161283810.32917/hmj/12061268770909.35036 KrutitskiiP. A.The Helmholtz equation in the exterior of slits in a plane with different impedance boundary conditions on opposite sides of the slitsQ. Appl. Math.2009677392249507210.1090/S0033-569X-08-01117-41163.35342 GuntherN. M.Potential Theory and Its Application to the Fundamental Problems of Mathematical Physics1967New YorkUngar KrutitskiiP. A.The skew derivative problem in the exterior of open curves in a planeZ. Anal. Anwend.199716739747147272810.4171/ZAA/7880888.35027 KrutitskiiP. A.Wave propagation in a 2-D external domain with cutsAppl. Anal.199662297309162350710.1080/000368196088404840857.73023 VladimirovV. S.Equations of Mathematical Physics1971New YorkMarcel Dekker0231.35002 KrutitskiiP. A.KwakD. Y.HyonY. K.Numerical treatment of a skew-derivative problem for the Laplace equation in the exterior of an open arcJ. Eng. Math.2007592560234276810.1007/s10665-006-9058-x1126.78018 KrutitskiiP. A.KolybasovaV. V.Numerical method for the solution of integral equations in a problem with directional derivative for the Laplace equation outside open curvesDiffer. Equations20165212191233356362610.1134/S00122661160901351357.65321 KrutitskiiP. A.On the electric current from electrodes in a magnetized semiconductor filmIMA J. Appl. Math.199860285297165386310.1093/imamat/60.3.2850915.35100 1771_CR6 P. A. Krutitskii (1771_CR15) 1996; 62 V. F. Butuzov (1771_CR17) 2000 P. A. Krutitskii (1771_CR13) 1998; 60 1771_CR1 P. A. Krutitskii (1771_CR11) 1998; 28 N. M. Gunther (1771_CR2) 1967 P. A. Krutitskii (1771_CR9) 1998; 17 P. A. Krutitskii (1771_CR14) 1998; 38 P. A. Krutitskii (1771_CR8) 2016; 52 P. A. Krutitskii (1771_CR12) 1998; 21 D. Colton (1771_CR3) 1984 P. A. Krutitskii (1771_CR10) 1997; 16 P. A. Krutitskii (1771_CR16) 2009; 67 V. S. Vladimirov (1771_CR19) 1971 P. A. Krutitskii (1771_CR5) 2019; 55 1771_CR18 A. P. Prudnikov (1771_CR20) 1986 C. A. Brebbia (1771_CR4) 1984 P. A. Krutitskii (1771_CR7) 2007; 59 |
| References_xml | – reference: A. N. Tikhonov and A. A. Samarskii, Equations of Mathematical Physics (Nauka, Moscow, 1999; Dover, New York, 2011). – reference: KrutitskiiP. A.The 2-dimensional Dirichlet problem in an external domain with cutsZ. Anal. Anwend.199817361378163255110.4171/ZAA/8270907.35036 – reference: KrutitskiiP. A.The Neumann problem on wave propagation in a 2-D external domain with cutsJ. Math. Kyoto Univ.19983843945216612040948.35031 – reference: BrebbiaC. A.TellesJ. C. F.WrobelL. C.Boundary Element Techniques: Theory and Applications in Engineering1984BerlinSpringer-Verlag10.1007/978-3-642-48860-30556.73086 – reference: KrutitskiiP. A.The Dirichlet problem for the dissipative Helmholtz equation in a plane domain bounded by closed and open curvesHiroshima Math. J.199828149168161283810.32917/hmj/12061268770909.35036 – reference: KrutitskiiP. A.KolybasovaV. V.Numerical method for the solution of integral equations in a problem with directional derivative for the Laplace equation outside open curvesDiffer. Equations20165212191233356362610.1134/S00122661160901351357.65321 – reference: GuntherN. M.Potential Theory and Its Application to the Fundamental Problems of Mathematical Physics1967New YorkUngar – reference: V. A. Il’in and E. G. Poznyak, Fundamentals of Calculus (Fizmatlit, Moscow, 1973), Part 2 [in Russian]. – reference: P. A. Krutitskii, A. D. Fedotov, and V. V. Kolybasova, Preprint No. 112, IPM RAN (Keldysh Inst. of Applied Mathematics, Russian Academy of Sciences, Moscow, 2019). – reference: ColtonD.KressR.Integral Equation Methods in Scattering Theory1984New YorkWiley0522.35001 – reference: KrutitskiiP. A.KwakD. Y.HyonY. K.Numerical treatment of a skew-derivative problem for the Laplace equation in the exterior of an open arcJ. Eng. Math.2007592560234276810.1007/s10665-006-9058-x1126.78018 – reference: VladimirovV. S.Equations of Mathematical Physics1971New YorkMarcel Dekker0231.35002 – reference: KrutitskiiP. A.The skew derivative problem in the exterior of open curves in a planeZ. Anal. Anwend.199716739747147272810.4171/ZAA/7880888.35027 – reference: KrutitskiiP. A.On the electric current from electrodes in a magnetized semiconductor filmIMA J. Appl. Math.199860285297165386310.1093/imamat/60.3.2850915.35100 – reference: KrutitskiiP. A.Wave propagation in a 2-D external domain with cutsAppl. Anal.199662297309162350710.1080/000368196088404840857.73023 – reference: KrutitskiiP. A.The Helmholtz equation in the exterior of slits in a plane with different impedance boundary conditions on opposite sides of the slitsQ. Appl. Math.2009677392249507210.1090/S0033-569X-08-01117-41163.35342 – reference: KrutitskiiP. A.FedotovA. D.KolybasovaV. V.Quadrature formula for the simple layer potentialDiffer. Equations20195512261241402804410.1134/S00122661190901061455.65040 – reference: ButuzovV. F.KrutitskayaN. Ch.MedvedevG. N.ShishkinA. A.Calculus in Questions and Problems2000MoscowFizmatlit1029.00002 – reference: KrutitskiiP. A.The Neumann problem for the 2-D Helmholtz equation in a domain, bounded by closed and open curvesInt. J. Math. Math. Sci.199821209216160970810.1155/S01611712980003010910.35040 – reference: PrudnikovA. P.BrychkovYu. A.MarichevO. I.Integrals and Series1986New YorkGordon and Breach0606.33001 – volume: 55 start-page: 1226 year: 2019 ident: 1771_CR5 publication-title: Differ. Equations doi: 10.1134/S0012266119090106 – volume: 28 start-page: 149 year: 1998 ident: 1771_CR11 publication-title: Hiroshima Math. J. doi: 10.32917/hmj/1206126877 – volume: 62 start-page: 297 year: 1996 ident: 1771_CR15 publication-title: Appl. Anal. doi: 10.1080/00036819608840484 – volume-title: Calculus in Questions and Problems year: 2000 ident: 1771_CR17 – ident: 1771_CR6 – volume: 52 start-page: 1219 year: 2016 ident: 1771_CR8 publication-title: Differ. 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Kyoto Univ. – volume: 16 start-page: 739 year: 1997 ident: 1771_CR10 publication-title: Z. Anal. Anwend. doi: 10.4171/ZAA/788 – ident: 1771_CR18 – volume-title: Integral Equation Methods in Scattering Theory year: 1984 ident: 1771_CR3 – volume: 67 start-page: 73 year: 2009 ident: 1771_CR16 publication-title: Q. Appl. Math. doi: 10.1090/S0033-569X-08-01117-4 – volume: 21 start-page: 209 year: 1998 ident: 1771_CR12 publication-title: Int. J. Math. Math. Sci. doi: 10.1155/S0161171298000301 |
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| SubjectTerms | Accuracy Boundary value problems Computational Mathematics and Numerical Analysis General Numerical Methods Helmholtz equations Integral equations Integrals Mathematical analysis Mathematics Mathematics and Statistics Quadratures |
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| Title | Improved Quadrature Formula for a Single-Layer Potential |
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