Trefftz approximations in complex media : accuracy and applications
Approximations by Trefftz functions are rapidly gaining popularity in the numerical solution of boundary value problems of mathematical physics. By definition, these functions satisfy locally, in weak form, the underlying differential equations of the problem, which often results in high-order or ev...
| Main Authors: | , , , |
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| Other Authors: | |
| Format: | Journal Article |
| Language: | English |
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2020
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| Online Access: | https://hdl.handle.net/10356/143656 |
| _version_ | 1811680619381391360 |
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| author | Tsukerman, Igor Mansha, Shampy Chong, Yidong Markel, Vadim A. |
| author2 | School of Physical and Mathematical Sciences |
| author_facet | School of Physical and Mathematical Sciences Tsukerman, Igor Mansha, Shampy Chong, Yidong Markel, Vadim A. |
| author_sort | Tsukerman, Igor |
| collection | NTU |
| description | Approximations by Trefftz functions are rapidly gaining popularity in the numerical solution of boundary value problems of mathematical physics. By definition, these functions satisfy locally, in weak form, the underlying differential equations of the problem, which often results in high-order or even exponential accuracy with respect to the size of the basis set. We highlight two separate examples in applied electromagnetics and photonics: (i) homogenization of periodic structures, and (ii) numerical simulation of electromagnetic waves in slab geometries. Extensive numerical evidence and theoretical considerations show that Trefftz approximations can be applied much more broadly than is traditionally done: they are effective not only in physically homogeneous regions but also in complex inhomogeneous ones. Two mechanisms underlying the high accuracy of Trefftz approximations in such complex cases are pointed out. The first one is related to trigonometric interpolation and the second one – somewhat surprisingly – to well-posedness of random matrices. |
| first_indexed | 2024-10-01T03:27:56Z |
| format | Journal Article |
| id | ntu-10356/143656 |
| institution | Nanyang Technological University |
| language | English |
| last_indexed | 2024-10-01T03:27:56Z |
| publishDate | 2020 |
| record_format | dspace |
| spelling | ntu-10356/1436562023-02-28T19:26:15Z Trefftz approximations in complex media : accuracy and applications Tsukerman, Igor Mansha, Shampy Chong, Yidong Markel, Vadim A. School of Physical and Mathematical Sciences Science::Mathematics Trefftz Approximations Maxwell’s Equations Approximations by Trefftz functions are rapidly gaining popularity in the numerical solution of boundary value problems of mathematical physics. By definition, these functions satisfy locally, in weak form, the underlying differential equations of the problem, which often results in high-order or even exponential accuracy with respect to the size of the basis set. We highlight two separate examples in applied electromagnetics and photonics: (i) homogenization of periodic structures, and (ii) numerical simulation of electromagnetic waves in slab geometries. Extensive numerical evidence and theoretical considerations show that Trefftz approximations can be applied much more broadly than is traditionally done: they are effective not only in physically homogeneous regions but also in complex inhomogeneous ones. Two mechanisms underlying the high accuracy of Trefftz approximations in such complex cases are pointed out. The first one is related to trigonometric interpolation and the second one – somewhat surprisingly – to well-posedness of random matrices. Ministry of Education (MOE) Accepted version The work of IT was supported in part by the US National Science Foundation Grants DMS-1216927 and DMS-1620112. There search of SM and YC was supported by the Singapore MOE Academic Research Fund Tier 2 Grant MOE 2016-T2-1-128, the Singapore MOE Academic Research Fund Tier 2 Grant MOE2015-T2-2-008, and the Singapore MOE Academic Research Fund Tier 3 Grant MOE 2016-T3-1-006. The work of VM was supported in part by the US National Science Foundation Grants DMS-1216970. IT thanks Ralf Hiptmair, Andrea Moiola, Lise-Marie Imbert-Gérard and Ben Schweizer for discussions. 2020-09-15T07:00:40Z 2020-09-15T07:00:40Z 2018 Journal Article Tsukerman, I., Mansha, S., Chong, Y., & Markel, V. A. (2019). Trefftz approximations in complex media : accuracy and applications. Computers and Mathematics with Applications, 77(6), 1770-1785. doi:10.1016/j.camwa.2018.08.065 0898-1221 https://hdl.handle.net/10356/143656 10.1016/j.camwa.2018.08.065 6 77 1770 1785 en Computers and Mathematics with Applications © 2018 Elsevier Ltd. All rights reserved. This paper was published in Computers and Mathematics with Applications and is made available with permission of Elsevier Ltd. application/pdf |
| spellingShingle | Science::Mathematics Trefftz Approximations Maxwell’s Equations Tsukerman, Igor Mansha, Shampy Chong, Yidong Markel, Vadim A. Trefftz approximations in complex media : accuracy and applications |
| title | Trefftz approximations in complex media : accuracy and applications |
| title_full | Trefftz approximations in complex media : accuracy and applications |
| title_fullStr | Trefftz approximations in complex media : accuracy and applications |
| title_full_unstemmed | Trefftz approximations in complex media : accuracy and applications |
| title_short | Trefftz approximations in complex media : accuracy and applications |
| title_sort | trefftz approximations in complex media accuracy and applications |
| topic | Science::Mathematics Trefftz Approximations Maxwell’s Equations |
| url | https://hdl.handle.net/10356/143656 |
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