A Hausdorff-measure boundary element method for acoustic scattering by fractal screens

Sound-soft fractal screens can scatter acoustic waves even when they have zero surface measure. To solve such scattering problems we make what appears to be the first application of the boundary element method (BEM) where each BEM basis function is supported in a fractal set, and the integration inv...

Celý popis

Uložené v:

Podrobná bibliografia
Vydané v:	Numerische Mathematik Ročník 156; číslo 2; s. 463 - 532
Hlavní autori:	Caetano, A. M., Chandler-Wilde, S. N., Gibbs, A., Hewett, D. P., Moiola, A.
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Berlin/Heidelberg Springer Berlin Heidelberg 01.04.2024 Springer Nature B.V
Predmet:	Acoustic scattering Acoustic waves Acoustics Basis functions Boundary element method Boundary integral method Convergence Estimates Fractals Function space Integral equations Mathematical analysis Mathematical and Computational Engineering Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Numerical Analysis Numerical and Computational Physics Regularity Screens Simulation Theoretical 45A05 28A80 46E35 65R20 65N38
ISSN:	0029-599X, 0945-3245
On-line prístup:	Získať plný text
Tagy:	Pridať tag Žiadne tagy, Buďte prvý, kto otaguje tento záznam!

Abstract	Sound-soft fractal screens can scatter acoustic waves even when they have zero surface measure. To solve such scattering problems we make what appears to be the first application of the boundary element method (BEM) where each BEM basis function is supported in a fractal set, and the integration involved in the formation of the BEM matrix is with respect to a non-integer order Hausdorff measure rather than the usual (Lebesgue) surface measure. Using recent results on function spaces on fractals, we prove convergence of the Galerkin formulation of this “Hausdorff BEM” for acoustic scattering in R n + 1 ( n = 1 , 2 ) when the scatterer, assumed to be a compact subset of R n × { 0 } , is a d -set for some d ∈ ( n - 1 , n ] , so that, in particular, the scatterer has Hausdorff dimension d . For a class of fractals that are attractors of iterated function systems, we prove convergence rates for the Hausdorff BEM and superconvergence for smooth antilinear functionals, under certain natural regularity assumptions on the solution of the underlying boundary integral equation. We also propose numerical quadrature routines for the implementation of our Hausdorff BEM, along with a fully discrete convergence analysis, via numerical (Hausdorff measure) integration estimates and inverse estimates on fractals, estimating the discrete condition numbers. Finally, we show numerical experiments that support the sharpness of our theoretical results, and our solution regularity assumptions, including results for scattering in R 2 by Cantor sets, and in R 3 by Cantor dusts.
AbstractList	Sound-soft fractal screens can scatter acoustic waves even when they have zero surface measure. To solve such scattering problems we make what appears to be the first application of the boundary element method (BEM) where each BEM basis function is supported in a fractal set, and the integration involved in the formation of the BEM matrix is with respect to a non-integer order Hausdorff measure rather than the usual (Lebesgue) surface measure. Using recent results on function spaces on fractals, we prove convergence of the Galerkin formulation of this “Hausdorff BEM” for acoustic scattering in $$\mathbb {R}^{n+1}$$ R n + 1 ( $$n=1,2$$ n = 1 , 2 ) when the scatterer, assumed to be a compact subset of $$\mathbb {R}^n\times \{0\}$$ R n × { 0 } , is a d -set for some $$d\in (n-1,n]$$ d ∈ ( n - 1 , n ] , so that, in particular, the scatterer has Hausdorff dimension d . For a class of fractals that are attractors of iterated function systems, we prove convergence rates for the Hausdorff BEM and superconvergence for smooth antilinear functionals, under certain natural regularity assumptions on the solution of the underlying boundary integral equation. We also propose numerical quadrature routines for the implementation of our Hausdorff BEM, along with a fully discrete convergence analysis, via numerical (Hausdorff measure) integration estimates and inverse estimates on fractals, estimating the discrete condition numbers. Finally, we show numerical experiments that support the sharpness of our theoretical results, and our solution regularity assumptions, including results for scattering in $$\mathbb {R}^2$$ R 2 by Cantor sets, and in $$\mathbb {R}^3$$ R 3 by Cantor dusts. Sound-soft fractal screens can scatter acoustic waves even when they have zero surface measure. To solve such scattering problems we make what appears to be the first application of the boundary element method (BEM) where each BEM basis function is supported in a fractal set, and the integration involved in the formation of the BEM matrix is with respect to a non-integer order Hausdorff measure rather than the usual (Lebesgue) surface measure. Using recent results on function spaces on fractals, we prove convergence of the Galerkin formulation of this “Hausdorff BEM” for acoustic scattering in Rn+1 (n=1,2) when the scatterer, assumed to be a compact subset of Rn×{0}, is a d-set for some d∈(n-1,n], so that, in particular, the scatterer has Hausdorff dimension d. For a class of fractals that are attractors of iterated function systems, we prove convergence rates for the Hausdorff BEM and superconvergence for smooth antilinear functionals, under certain natural regularity assumptions on the solution of the underlying boundary integral equation. We also propose numerical quadrature routines for the implementation of our Hausdorff BEM, along with a fully discrete convergence analysis, via numerical (Hausdorff measure) integration estimates and inverse estimates on fractals, estimating the discrete condition numbers. Finally, we show numerical experiments that support the sharpness of our theoretical results, and our solution regularity assumptions, including results for scattering in R2 by Cantor sets, and in R3 by Cantor dusts. Sound-soft fractal screens can scatter acoustic waves even when they have zero surface measure. To solve such scattering problems we make what appears to be the first application of the boundary element method (BEM) where each BEM basis function is supported in a fractal set, and the integration involved in the formation of the BEM matrix is with respect to a non-integer order Hausdorff measure rather than the usual (Lebesgue) surface measure. Using recent results on function spaces on fractals, we prove convergence of the Galerkin formulation of this “Hausdorff BEM” for acoustic scattering in R n + 1 ( n = 1 , 2 ) when the scatterer, assumed to be a compact subset of R n × { 0 } , is a d -set for some d ∈ ( n - 1 , n ] , so that, in particular, the scatterer has Hausdorff dimension d . For a class of fractals that are attractors of iterated function systems, we prove convergence rates for the Hausdorff BEM and superconvergence for smooth antilinear functionals, under certain natural regularity assumptions on the solution of the underlying boundary integral equation. We also propose numerical quadrature routines for the implementation of our Hausdorff BEM, along with a fully discrete convergence analysis, via numerical (Hausdorff measure) integration estimates and inverse estimates on fractals, estimating the discrete condition numbers. Finally, we show numerical experiments that support the sharpness of our theoretical results, and our solution regularity assumptions, including results for scattering in R 2 by Cantor sets, and in R 3 by Cantor dusts.
Author	Hewett, D. P. Chandler-Wilde, S. N. Caetano, A. M. Gibbs, A. Moiola, A.
Author_xml	– sequence: 1 givenname: A. M. surname: Caetano fullname: Caetano, A. M. organization: Center for R &D in Mathematics and Applications, Departamento de Matemática, Universidade de Aveiro – sequence: 2 givenname: S. N. surname: Chandler-Wilde fullname: Chandler-Wilde, S. N. organization: Department of Mathematics and Statistics, University of Reading – sequence: 3 givenname: A. surname: Gibbs fullname: Gibbs, A. organization: Department of Mathematics, University College London – sequence: 4 givenname: D. P. surname: Hewett fullname: Hewett, D. P. email: d.hewett@ucl.ac.uk organization: Department of Mathematics, University College London – sequence: 5 givenname: A. surname: Moiola fullname: Moiola, A. organization: Dipartimento di Matematica “F. Casorati”, Università degli studi di Pavia
BookMark	eNp9kEtLAzEQgINUsK3-AU8Bz9G8Nrs5lqJWKHhR8Ray2YluaTc1yR767926guChpxmG-ebxzdCkCx0gdM3oLaO0vEuUcsYI5ZJQJrQm5RmaUi0LIrgsJkNOuSaF1u8XaJbShlJWKsmm6G2BV7ZPTYjekx3Y1EfAdei7xsYDhi3soMt4B_kzNNiHiK0Lfcqtw8nZnCG23QeuD9hH67LdDtUI0KVLdO7tNsHVb5yj14f7l-WKrJ8fn5aLNXGi0pkUFa8ZVYKLpnK1F5Kpxtal9dp7UBVtFFCruNTSaSs8F3VdDu86xYGXRW3FHN2Mc_cxfPWQstmEPnbDSiOo5FKoiqmhqxq7XAwpRfDGtdnmNnQ52nZrGDVHi2a0aAaL5seiKQeU_0P3sd0Nbk5DYoTS_ugH4t9VJ6hvQhSHaw
CitedBy_id	crossref_primary_10_1515_gmj_2025_2067 crossref_primary_10_1007_s11785_025_01717_3
Cites_doi	10.1007/BF02476031 10.1016/S0045-7949(96)00137-X 10.1016/j.jfa.2021.109019 10.1080/03605309108820799 10.1002/mma.1670130403 10.1007/s00211-021-01189-5 10.1142/S0218202522500075 10.1017/CBO9780511618635 10.1007/s00211-021-01182-y 10.1007/BF01396766 10.1007/978-0-387-09432-8 10.1006/jfan.1998.3383 10.1006/jcph.1994.1116 10.1137/17M1131933 10.1007/978-3-642-66451-9 10.1080/17476933.2011.557154 10.1080/00036818408839520 10.1017/S0013091500019076 10.1007/s10543-021-00859-y 10.1007/s00041-011-9202-5 10.1017/CBO9780511623738 10.1093/imanum/drab074 10.1007/s11075-022-01378-9 10.1142/S021953051650024X 10.1007/978-0-387-68805-3 10.1090/S0025-5718-03-01583-7 10.1093/imanum/8.1.105 10.1016/j.jmaa.2004.10.059 10.1017/S0962492912000037 10.1007/978-3-540-68093-2 10.1115/1.3424474 10.1007/s00020-017-2342-5 10.1007/978-3-0348-0034-1 10.1109/MAP.2003.1189650 10.1007/s11075-023-01705-8 10.4171/019 10.1016/j.jmaa.2019.01.036 10.1007/BF01199079 10.1112/mtk.12155 10.1080/02786820117868 10.1090/gsm/019 10.1007/s00020-015-2233-6 10.1112/S0025579314000278
ContentType	Journal Article
Copyright	The Author(s) 2024 The Author(s) 2024. This work is published under http://creativecommons.org/licenses/by/4.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.
Copyright_xml	– notice: The Author(s) 2024 – notice: The Author(s) 2024. This work is published under http://creativecommons.org/licenses/by/4.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.
DBID	C6C AAYXX CITATION
DOI	10.1007/s00211-024-01399-7
DatabaseName	Springer Nature OA Free Journals CrossRef
DatabaseTitle	CrossRef
DatabaseTitleList	CrossRef
DeliveryMethod	fulltext_linktorsrc
Discipline	Mathematics Computer Science
EISSN	0945-3245
EndPage	532
ExternalDocumentID	10_1007_s00211_024_01399_7
GroupedDBID	--Z -52 -5D -5G -BR -EM -Y2 -~C -~X .86 .DC .VR 06D 0R~ 0VY 123 199 1N0 1SB 203 29N 2J2 2JN 2JY 2KG 2KM 2LR 2P1 2VQ 2WC 2~H 30V 4.4 406 408 409 40D 40E 41~ 5QI 5VS 67Z 692 6NX 6TJ 78A 8TC 8UJ 95- 95. 95~ 96X AAAVM AABHQ AACDK AAHNG AAIAL AAJBT AAJKR AANZL AARHV AARTL AASML AATNV AATVU AAUYE AAWCG AAYIU AAYQN AAYTO AAYZH ABAKF ABBBX ABBXA ABDZT ABECU ABFTV ABHLI ABHQN ABJNI ABJOX ABKCH ABKTR ABLJU ABMNI ABMQK ABNWP ABQBU ABQSL ABSXP ABTEG ABTHY ABTKH ABTMW ABULA ABWNU ABXPI ACAOD ACBXY ACGFO ACGFS ACHSB ACHXU ACIWK ACKNC ACMDZ ACMLO ACOKC ACOMO ACPIV ACREN ACZOJ ADHHG ADHIR ADIMF ADINQ ADKNI ADKPE ADRFC ADTPH ADURQ ADYFF ADYOE ADZKW AEBTG AEFIE AEFQL AEGAL AEGNC AEJHL AEJRE AEKMD AEMSY AENEX AEOHA AEPYU AESKC AETLH AEVLU AEXYK AFBBN AFDYV AFEXP AFFNX AFGCZ AFLOW AFQWF AFWTZ AFYQB AFZKB AGAYW AGDGC AGGDS AGJBK AGMZJ AGQEE AGQMX AGRTI AGWIL AGWZB AGYKE AHAVH AHBYD AHKAY AHSBF AHYZX AIAKS AIGIU AIIXL AILAN AITGF AJBLW AJRNO AJZVZ ALMA_UNASSIGNED_HOLDINGS ALWAN AMKLP AMTXH AMXSW AMYLF AMYQR AOCGG ARMRJ ASPBG AVWKF AXYYD AYJHY AZFZN B-. BA0 BAPOH BBWZM BDATZ BGNMA BSONS C6C CAG COF CS3 CSCUP DDRTE DL5 DNIVK DPUIP EBLON EBS EIOEI EJD ESBYG FEDTE FERAY FFXSO FIGPU FINBP FNLPD FRRFC FSGXE FWDCC GGCAI GGRSB GJIRD GNWQR GQ6 GQ7 GQ8 GXS H13 HF~ HG5 HG6 HMJXF HQYDN HRMNR HVGLF HZ~ H~9 I09 IHE IJ- IKXTQ ITM IWAJR IXC IZIGR IZQ I~X I~Z J-C J0Z JBSCW JCJTX JZLTJ KDC KOV KOW KQ8 LAS LLZTM LO0 M4Y MA- N2Q N9A NB0 NDZJH NPVJJ NQJWS NU0 O9- O93 O9G O9I O9J OAM OK1 P19 P2P P9R PF0 PKN PT4 PT5 QOK QOS R4E R89 R9I REI RHV RIG RNI RNS ROL RPX RSV RYB RZK RZZ S16 S1Z S26 S27 S28 S3B SAP SCLPG SDD SDH SDM SHX SISQX SJYHP SMT SNE SNPRN SNX SOHCF SOJ SPISZ SRMVM SSLCW STPWE SZN T13 T16 TN5 TSG TSK TSV TUC U2A UG4 UOJIU UTJUX UZXMN VC2 VFIZW W23 W48 WH7 WK8 XJT YLTOR YNT YQT Z45 Z5O Z7R Z7X Z83 Z86 Z88 Z8M Z8R Z8W Z92 ZMTXR ZWQNP ~EX AAPKM AAYXX ABBRH ABDBE ABFSG ABRTQ ABUFD ACSTC ADHKG AEZWR AFDZB AFHIU AFOHR AGQPQ AHPBZ AHWEU AIXLP ATHPR AYFIA CITATION
ID	FETCH-LOGICAL-c389t-582b106323d8cbf3416dab7af9ffe680d6e0a62494c9a3f23bb7007c62e275ba3
IEDL.DBID	RSV
ISICitedReferencesCount	3
ISICitedReferencesURI	http://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=Summon&SrcAuth=ProQuest&DestLinkType=CitingArticles&DestApp=WOS_CPL&KeyUT=001172025600001&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D
ISSN	0029-599X
IngestDate	Mon Oct 06 16:24:24 EDT 2025 Sat Nov 29 01:36:01 EST 2025 Tue Nov 18 22:16:38 EST 2025 Fri Feb 21 02:40:37 EST 2025
IsDoiOpenAccess	true
IsOpenAccess	true
IsPeerReviewed	true
IsScholarly	true
Issue	2
Keywords	45A05 28A80 46E35 65R20 65N38
Language	English
LinkModel	DirectLink
MergedId	FETCHMERGED-LOGICAL-c389t-582b106323d8cbf3416dab7af9ffe680d6e0a62494c9a3f23bb7007c62e275ba3
Notes	ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14
OpenAccessLink	https://link.springer.com/10.1007/s00211-024-01399-7
PQID	3042436816
PQPubID	2043616
PageCount	70
ParticipantIDs	proquest_journals_3042436816 crossref_citationtrail_10_1007_s00211_024_01399_7 crossref_primary_10_1007_s00211_024_01399_7 springer_journals_10_1007_s00211_024_01399_7
PublicationCentury	2000
PublicationDate	2024-04-01
PublicationDateYYYYMMDD	2024-04-01
PublicationDate_xml	– month: 04 year: 2024 text: 2024-04-01 day: 01
PublicationDecade	2020
PublicationPlace	Berlin/Heidelberg
PublicationPlace_xml	– name: Berlin/Heidelberg – name: Heidelberg
PublicationTitle	Numerische Mathematik
PublicationTitleAbbrev	Numer. Math
PublicationYear	2024
Publisher	Springer Berlin Heidelberg Springer Nature B.V
Publisher_xml	– name: Springer Berlin Heidelberg – name: Springer Nature B.V
References	Riddle, L.: Classic Iterated Function Systems. larryriddle.agnesscott.org/ifs/ifs.htm, downloaded 18 July 2023 SloanIHSpenceAThe Galerkin method for integral equations of the first kind with logarithmic kernel: TheoryIMA J. Numer. Anal.1988810512296784610.1093/imanum/8.1.105 Chandler-WildeSNHewettDPMoiolaABessonJBoundary element methods for acoustic scattering by fractal screensNumer. Math.2021147785837424546210.1007/s00211-021-01182-y Falconer, K.: Fractal Geometry: Mathematical Foundations and Applications, 3rd edn. Wiley (2014) HewettDPMoiolaAOn the maximal Sobolev regularity of distributions supported by subsets of Euclidean spaceAnal. Appl.201715731770367037810.1142/S021953051650024X JonssonAWavelets on fractals and Besov spacesJ. Fourier Anal. Appl.19984329340165098010.1007/BF02476031 SorensenCMLight scattering by fractal aggregates: a reviewAerosol Sci. Tech.20013564868710.1080/02786820117868 CwikelMKaltonNJInterpolation of compact operators by the methods of Calderón and Gustavsson–PeetreProc. Edin. Math. Soc.19953826127610.1017/S0013091500019076 Grafakos, L.: Classical Fourier Analysis. Springer (2008) ClaeysXGiacomelLHiptmairRUrzúa-TorresCQuotient-space boundary element methods for scattering at complex screensBIT Numer. Math.20216111931221433213510.1007/s10543-021-00859-y GibbsAHewettDPMoiolaANumerical evaluation of singular integrals on fractal setsNumer. Alg.2023922071212410.1007/s11075-022-01378-9 Chandler-WildeSNHewettDPMoiolaASobolev spaces on non-Lipschitz subsets of Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{R} ^n$$\end{document} with application to boundary integral equations on fractal screensIntegr. Equat. Operat. Th.20178717922410.1007/s00020-017-2342-5 Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland (1978) Steinbach, O.: Numerical Approximation Methods for Elliptic Boundary Value Problems. Springer (2008) AlougesFAversengMNew preconditioners for the Laplace and Helmholtz integral equations on open curves: analytical framework and numerical resultsNumer. Math.2021148255292427947510.1007/s00211-021-01189-5 ZubermanLExact Hausdorff and packing measure of certain Cantor sets, not necessarily self-similar or homogeneousJ. Math. Anal. Appl.2019474143156391289510.1016/j.jmaa.2019.01.036 CostabelMStephanEPDuality estimates for the numerical solution of integral equationsNumer. Math.19885433935397170710.1007/BF01396766 DahmenWFaermannBGrahamIGHackbuschWSauterSAInverse inequalities on non-quasi-uniform meshes and application to the mortar element methodMath. Comp.20047311071138204708010.1090/S0025-5718-03-01583-7 Falconer, K.: The Geometry of Fractal Sets. CUP (1985) Jonsson, A., Wallin, H.: Function Spaces on Subsets of Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}}^n$$\end{document}. Math. Rep. 2 (1984) ErvinVJ StephanEPAbou El-SeoudSAn improved boundary element method for the charge density of a thin electrified plate in R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{R} ^3$$\end{document}Math. Meth. Appl. Sci.19901329130310.1002/mma.1670130403 Gibbs, A., Hewett, D.P., Major, B.: Numerical evaluation of singular integrals on non-disjoint self-similar fractal sets. arXiv:2303.13141, (2023) MitreaMTaylorMBoundary layer methods for Lipschitz domains in Riemannian manifoldsJ. Funct. Anal.1999163181251168048710.1006/jfan.1998.3383 WernerDHGangulySAn overview of fractal antenna engineering researchIEEE Ant. Propag. Mag.200345385710.1109/MAP.2003.1189650 Evans, L.C.: Partial Differential Equations. AMS (2010) Sauter, S.A., Schwab, C.: Boundary Element Methods. Springer (2011) CaetanoAHewettDPMoiolaADensity results for Sobolev, Besov and Triebel–Lizorkin spaces on rough setsJ. Funct. Anal.2021281424370510.1016/j.jfa.2021.109019 Braess, D.: Finite Elements, third edn. CUP (2007) Caetano, A.M., Chandler-Wilde, S.N., Claeys, X., Gibbs, A., Hewett, D.P., Moiola, A.: Integral equation methods for acoustic scattering by fractals. arXiv:2309.02184 (2023) StephanEPWendlandWLAn augmented Galerkin procedure for the boundary integral method applied to two-dimensional screen and crack problemsAppl. Anal.19841818321976750010.1080/00036818408839520 StephanEPBoundary integral equations for screen problems in R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{R} ^3$$\end{document}Integr. Equat. Oper. Th.19871023625710.1007/BF01199079 Jerez-HanckesCPintoJSpectral Galerkin method for solving Helmholtz boundary integral equations on smooth screensIMA J. Numer. Anal.20224235713608449782410.1093/imanum/drab074 Chandler-WildeSNGrahamIGLangdonSSpenceEANumerical-asymptotic boundary integral methods in high-frequency acoustic scatteringActa Numer20122189305291638210.1017/S0962492912000037 Chandler-WildeSNHewettDPWell-posed PDE and integral equation formulations for scattering by fractal screensSIAM J. Math. Anal.201850677717375710010.1137/17M1131933 McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. CUP (2000) BannisterJGibbsAHewettDPAcoustic scattering by impedance screens/cracks with fractal boundary: well-posedness analysis and boundary element approximationMath. Mod. Meth. Appl. Sci.202232291319439615710.1142/S0218202522500075 ŠneıbergIJSpectral properties of linear operators in interpolation families of Banach spacesMat. Issled19749214229634681 XiongYZhouJThe Hausdorff measure of a class of Sierpinski carpetsJ. Math. Anal. Appl.2005305121129212811610.1016/j.jmaa.2004.10.059 Bricchi, M.: Tailored Function Spaces and h\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h$$\end{document}-sets. PhD thesis, University of Jena (2002) CaetanoAOn the type of convergence in atomic representationsComplex Var. Elliptic Equat.201156875883283822610.1080/17476933.2011.557154 Chandler-WildeSNHewettDPWavenumber-explicit continuity and coercivity estimates in acoustic scattering by planar screensIntegr. Equat. Oper. Th.201582423449335578810.1007/s00020-015-2233-6 PanagiotopoulosP PanagouliOThe FEM and BEM for fractal boundaries and interfaces. Applications to unilateral problemsComput. Struct.19976432933910.1016/S0045-7949(96)00137-X Chandler-WildeSNHewettDPMoiolaAInterpolation of Hilbert and Sobolev spaces: quantitative estimates and counterexamplesMathematika201561414443334306110.1112/S0025579314000278 HsiaoGCWendlandWLThe Aubin-Nitsche lemma for integral equationsJ. Integr. Equat.19813299315634453 Triebel, H.: Function Spaces and Wavelets on Domains. European Mathematical Society (2008) Caetano, A.M., Chandler-Wilde, S.N., Gibbs, A., Hewett, D.P.: Properties of IFS attractors with non-empty interiors and associated function spaces and scattering problems. (In preparation) Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover (1964) Chandler-Wilde, S.N., Hewett, D.P., Moiola, A., Corrigendum: Interpolation of Hilbert and Sobolev spaces: quantitative estimates and counterexamples (Mathematika 61, 414–443 (2015)). Mathematika 68, 1393–1400 (2022) BerghJLöfströmJInterpolation Spaces: An Introduction1976BerlinSpringer10.1007/978-3-642-66451-9 Bezanson, J., Karpinski, S., Shah, V. B., Edelman, A.: Julia: A fast dynamic language for technical computing. arXiv:1209.5145 (2012) JonesPMaJRokhlinVA fast direct algorithm for the solution of the Laplace equation on regions with fractal boundariesJ. Comput. Phys.19941133551127818910.1006/jcph.1994.1116 SchneiderRReduction of order for pseudodifferential operators on Lipschitz domainsCommun. Partial Differ. Equ.19911612631286113278510.1080/03605309108820799 Triebel, H.: Fractals and Spectra. Birkhäuser (1997) CarvalhoACaetanoAOn the Hausdorff dimension of continuous functions belonging to Hölder and Besov spaces on fractal d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d$$\end{document} -setsJ. Fourier Anal. Appl.201218386409289873410.1007/s00041-011-9202-5 EP Stephan (1399_CR49) 1984; 18 M Mitrea (1399_CR39) 1999; 163 1399_CR29 1399_CR28 1399_CR27 1399_CR26 A Caetano (1399_CR9) 2021; 281 VJ Ervin (1399_CR25) 1990; 13 DP Hewett (1399_CR32) 2017; 15 C Jerez-Hanckes (1399_CR34) 2022; 42 1399_CR20 1399_CR7 1399_CR6 1399_CR5 1399_CR38 1399_CR37 A Caetano (1399_CR8) 2011; 56 SN Chandler-Wilde (1399_CR19) 2021; 147 A Carvalho (1399_CR12) 2012; 18 1399_CR1 J Bannister (1399_CR3) 2022; 32 DH Werner (1399_CR52) 2003; 45 L Zuberman (1399_CR54) 2019; 474 M Costabel (1399_CR22) 1988; 54 CM Sorensen (1399_CR46) 2001; 35 F Alouges (1399_CR2) 2021; 148 1399_CR31 SN Chandler-Wilde (1399_CR14) 2015; 82 GC Hsiao (1399_CR33) 1981; 3 A Jonsson (1399_CR36) 1998; 4 W Dahmen (1399_CR24) 2004; 73 SN Chandler-Wilde (1399_CR16) 2015; 61 IH Sloan (1399_CR44) 1988; 8 R Schneider (1399_CR43) 1991; 16 1399_CR47 IJ Šneıberg (1399_CR45) 1974; 9 J Bergh (1399_CR4) 1976 1399_CR42 M Cwikel (1399_CR23) 1995; 38 1399_CR41 SN Chandler-Wilde (1399_CR17) 2017; 87 1399_CR18 X Claeys (1399_CR21) 2021; 61 P Jones (1399_CR35) 1994; 113 SN Chandler-Wilde (1399_CR13) 2012; 21 1399_CR50 EP Stephan (1399_CR48) 1987; 10 Y Xiong (1399_CR53) 2005; 305 1399_CR11 P Panagiotopoulos (1399_CR40) 1997; 64 1399_CR10 SN Chandler-Wilde (1399_CR15) 2018; 50 A Gibbs (1399_CR30) 2023; 92 1399_CR51
References_xml	– reference: Chandler-WildeSNHewettDPMoiolaASobolev spaces on non-Lipschitz subsets of Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{R} ^n$$\end{document} with application to boundary integral equations on fractal screensIntegr. Equat. Operat. Th.20178717922410.1007/s00020-017-2342-5 – reference: Jerez-HanckesCPintoJSpectral Galerkin method for solving Helmholtz boundary integral equations on smooth screensIMA J. Numer. Anal.20224235713608449782410.1093/imanum/drab074 – reference: ZubermanLExact Hausdorff and packing measure of certain Cantor sets, not necessarily self-similar or homogeneousJ. Math. Anal. Appl.2019474143156391289510.1016/j.jmaa.2019.01.036 – reference: JonesPMaJRokhlinVA fast direct algorithm for the solution of the Laplace equation on regions with fractal boundariesJ. Comput. Phys.19941133551127818910.1006/jcph.1994.1116 – reference: Evans, L.C.: Partial Differential Equations. AMS (2010) – reference: StephanEPBoundary integral equations for screen problems in R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{R} ^3$$\end{document}Integr. Equat. Oper. Th.19871023625710.1007/BF01199079 – reference: Chandler-WildeSNHewettDPMoiolaAInterpolation of Hilbert and Sobolev spaces: quantitative estimates and counterexamplesMathematika201561414443334306110.1112/S0025579314000278 – reference: Chandler-Wilde, S.N., Hewett, D.P., Moiola, A., Corrigendum: Interpolation of Hilbert and Sobolev spaces: quantitative estimates and counterexamples (Mathematika 61, 414–443 (2015)). Mathematika 68, 1393–1400 (2022) – reference: Riddle, L.: Classic Iterated Function Systems. larryriddle.agnesscott.org/ifs/ifs.htm, downloaded 18 July 2023 – reference: DahmenWFaermannBGrahamIGHackbuschWSauterSAInverse inequalities on non-quasi-uniform meshes and application to the mortar element methodMath. Comp.20047311071138204708010.1090/S0025-5718-03-01583-7 – reference: Bezanson, J., Karpinski, S., Shah, V. B., Edelman, A.: Julia: A fast dynamic language for technical computing. arXiv:1209.5145 (2012) – reference: MitreaMTaylorMBoundary layer methods for Lipschitz domains in Riemannian manifoldsJ. Funct. Anal.1999163181251168048710.1006/jfan.1998.3383 – reference: SchneiderRReduction of order for pseudodifferential operators on Lipschitz domainsCommun. Partial Differ. Equ.19911612631286113278510.1080/03605309108820799 – reference: CaetanoAOn the type of convergence in atomic representationsComplex Var. Elliptic Equat.201156875883283822610.1080/17476933.2011.557154 – reference: XiongYZhouJThe Hausdorff measure of a class of Sierpinski carpetsJ. Math. Anal. Appl.2005305121129212811610.1016/j.jmaa.2004.10.059 – reference: Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover (1964) – reference: CwikelMKaltonNJInterpolation of compact operators by the methods of Calderón and Gustavsson–PeetreProc. Edin. Math. Soc.19953826127610.1017/S0013091500019076 – reference: PanagiotopoulosP PanagouliOThe FEM and BEM for fractal boundaries and interfaces. Applications to unilateral problemsComput. Struct.19976432933910.1016/S0045-7949(96)00137-X – reference: CarvalhoACaetanoAOn the Hausdorff dimension of continuous functions belonging to Hölder and Besov spaces on fractal d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d$$\end{document} -setsJ. Fourier Anal. Appl.201218386409289873410.1007/s00041-011-9202-5 – reference: Chandler-WildeSNGrahamIGLangdonSSpenceEANumerical-asymptotic boundary integral methods in high-frequency acoustic scatteringActa Numer20122189305291638210.1017/S0962492912000037 – reference: Jonsson, A., Wallin, H.: Function Spaces on Subsets of Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}}^n$$\end{document}. Math. Rep. 2 (1984) – reference: BannisterJGibbsAHewettDPAcoustic scattering by impedance screens/cracks with fractal boundary: well-posedness analysis and boundary element approximationMath. Mod. Meth. Appl. Sci.202232291319439615710.1142/S0218202522500075 – reference: WernerDHGangulySAn overview of fractal antenna engineering researchIEEE Ant. Propag. Mag.200345385710.1109/MAP.2003.1189650 – reference: Gibbs, A., Hewett, D.P., Major, B.: Numerical evaluation of singular integrals on non-disjoint self-similar fractal sets. arXiv:2303.13141, (2023) – reference: Falconer, K.: The Geometry of Fractal Sets. CUP (1985) – reference: Caetano, A.M., Chandler-Wilde, S.N., Claeys, X., Gibbs, A., Hewett, D.P., Moiola, A.: Integral equation methods for acoustic scattering by fractals. arXiv:2309.02184 (2023) – reference: Braess, D.: Finite Elements, third edn. CUP (2007) – reference: CaetanoAHewettDPMoiolaADensity results for Sobolev, Besov and Triebel–Lizorkin spaces on rough setsJ. Funct. Anal.2021281424370510.1016/j.jfa.2021.109019 – reference: ŠneıbergIJSpectral properties of linear operators in interpolation families of Banach spacesMat. Issled19749214229634681 – reference: Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland (1978) – reference: ErvinVJ StephanEPAbou El-SeoudSAn improved boundary element method for the charge density of a thin electrified plate in R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{R} ^3$$\end{document}Math. Meth. Appl. Sci.19901329130310.1002/mma.1670130403 – reference: Falconer, K.: Fractal Geometry: Mathematical Foundations and Applications, 3rd edn. Wiley (2014) – reference: ClaeysXGiacomelLHiptmairRUrzúa-TorresCQuotient-space boundary element methods for scattering at complex screensBIT Numer. Math.20216111931221433213510.1007/s10543-021-00859-y – reference: JonssonAWavelets on fractals and Besov spacesJ. Fourier Anal. Appl.19984329340165098010.1007/BF02476031 – reference: Triebel, H.: Function Spaces and Wavelets on Domains. European Mathematical Society (2008) – reference: Chandler-WildeSNHewettDPMoiolaABessonJBoundary element methods for acoustic scattering by fractal screensNumer. Math.2021147785837424546210.1007/s00211-021-01182-y – reference: Steinbach, O.: Numerical Approximation Methods for Elliptic Boundary Value Problems. Springer (2008) – reference: CostabelMStephanEPDuality estimates for the numerical solution of integral equationsNumer. Math.19885433935397170710.1007/BF01396766 – reference: AlougesFAversengMNew preconditioners for the Laplace and Helmholtz integral equations on open curves: analytical framework and numerical resultsNumer. Math.2021148255292427947510.1007/s00211-021-01189-5 – reference: Bricchi, M.: Tailored Function Spaces and h\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h$$\end{document}-sets. PhD thesis, University of Jena (2002) – reference: Sauter, S.A., Schwab, C.: Boundary Element Methods. Springer (2011) – reference: Triebel, H.: Fractals and Spectra. Birkhäuser (1997) – reference: StephanEPWendlandWLAn augmented Galerkin procedure for the boundary integral method applied to two-dimensional screen and crack problemsAppl. Anal.19841818321976750010.1080/00036818408839520 – reference: Chandler-WildeSNHewettDPWavenumber-explicit continuity and coercivity estimates in acoustic scattering by planar screensIntegr. Equat. Oper. Th.201582423449335578810.1007/s00020-015-2233-6 – reference: BerghJLöfströmJInterpolation Spaces: An Introduction1976BerlinSpringer10.1007/978-3-642-66451-9 – reference: Caetano, A.M., Chandler-Wilde, S.N., Gibbs, A., Hewett, D.P.: Properties of IFS attractors with non-empty interiors and associated function spaces and scattering problems. (In preparation) – reference: HewettDPMoiolaAOn the maximal Sobolev regularity of distributions supported by subsets of Euclidean spaceAnal. Appl.201715731770367037810.1142/S021953051650024X – reference: McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. CUP (2000) – reference: HsiaoGCWendlandWLThe Aubin-Nitsche lemma for integral equationsJ. Integr. Equat.19813299315634453 – reference: Grafakos, L.: Classical Fourier Analysis. Springer (2008) – reference: SorensenCMLight scattering by fractal aggregates: a reviewAerosol Sci. Tech.20013564868710.1080/02786820117868 – reference: SloanIHSpenceAThe Galerkin method for integral equations of the first kind with logarithmic kernel: TheoryIMA J. Numer. Anal.1988810512296784610.1093/imanum/8.1.105 – reference: Chandler-WildeSNHewettDPWell-posed PDE and integral equation formulations for scattering by fractal screensSIAM J. Math. Anal.201850677717375710010.1137/17M1131933 – reference: GibbsAHewettDPMoiolaANumerical evaluation of singular integrals on fractal setsNumer. Alg.2023922071212410.1007/s11075-022-01378-9 – volume: 4 start-page: 329 year: 1998 ident: 1399_CR36 publication-title: J. Fourier Anal. Appl. doi: 10.1007/BF02476031 – volume: 64 start-page: 329 year: 1997 ident: 1399_CR40 publication-title: Comput. Struct. doi: 10.1016/S0045-7949(96)00137-X – ident: 1399_CR41 – ident: 1399_CR5 – volume: 9 start-page: 214 year: 1974 ident: 1399_CR45 publication-title: Mat. Issled – ident: 1399_CR1 – volume: 281 year: 2021 ident: 1399_CR9 publication-title: J. Funct. Anal. doi: 10.1016/j.jfa.2021.109019 – volume: 16 start-page: 1263 year: 1991 ident: 1399_CR43 publication-title: Commun. Partial Differ. Equ. doi: 10.1080/03605309108820799 – volume: 13 start-page: 291 year: 1990 ident: 1399_CR25 publication-title: Math. Meth. Appl. Sci. doi: 10.1002/mma.1670130403 – volume: 148 start-page: 255 year: 2021 ident: 1399_CR2 publication-title: Numer. Math. doi: 10.1007/s00211-021-01189-5 – volume: 32 start-page: 291 year: 2022 ident: 1399_CR3 publication-title: Math. Mod. Meth. Appl. Sci. doi: 10.1142/S0218202522500075 – ident: 1399_CR28 – ident: 1399_CR6 doi: 10.1017/CBO9780511618635 – volume: 147 start-page: 785 year: 2021 ident: 1399_CR19 publication-title: Numer. Math. doi: 10.1007/s00211-021-01182-y – volume: 54 start-page: 339 year: 1988 ident: 1399_CR22 publication-title: Numer. Math. doi: 10.1007/BF01396766 – ident: 1399_CR31 doi: 10.1007/978-0-387-09432-8 – volume: 163 start-page: 181 year: 1999 ident: 1399_CR39 publication-title: J. Funct. Anal. doi: 10.1006/jfan.1998.3383 – volume: 113 start-page: 35 year: 1994 ident: 1399_CR35 publication-title: J. Comput. Phys. doi: 10.1006/jcph.1994.1116 – volume: 50 start-page: 677 year: 2018 ident: 1399_CR15 publication-title: SIAM J. Math. Anal. doi: 10.1137/17M1131933 – volume: 3 start-page: 299 year: 1981 ident: 1399_CR33 publication-title: J. Integr. Equat. – ident: 1399_CR38 – volume-title: Interpolation Spaces: An Introduction year: 1976 ident: 1399_CR4 doi: 10.1007/978-3-642-66451-9 – volume: 56 start-page: 875 year: 2011 ident: 1399_CR8 publication-title: Complex Var. Elliptic Equat. doi: 10.1080/17476933.2011.557154 – volume: 18 start-page: 183 year: 1984 ident: 1399_CR49 publication-title: Appl. Anal. doi: 10.1080/00036818408839520 – volume: 38 start-page: 261 year: 1995 ident: 1399_CR23 publication-title: Proc. Edin. Math. Soc. doi: 10.1017/S0013091500019076 – volume: 61 start-page: 1193 year: 2021 ident: 1399_CR21 publication-title: BIT Numer. Math. doi: 10.1007/s10543-021-00859-y – volume: 18 start-page: 386 year: 2012 ident: 1399_CR12 publication-title: J. Fourier Anal. Appl. doi: 10.1007/s00041-011-9202-5 – ident: 1399_CR27 doi: 10.1017/CBO9780511623738 – volume: 42 start-page: 3571 year: 2022 ident: 1399_CR34 publication-title: IMA J. Numer. Anal. doi: 10.1093/imanum/drab074 – ident: 1399_CR7 – volume: 92 start-page: 2071 year: 2023 ident: 1399_CR30 publication-title: Numer. Alg. doi: 10.1007/s11075-022-01378-9 – volume: 15 start-page: 731 year: 2017 ident: 1399_CR32 publication-title: Anal. Appl. doi: 10.1142/S021953051650024X – ident: 1399_CR37 – ident: 1399_CR10 – ident: 1399_CR47 doi: 10.1007/978-0-387-68805-3 – volume: 73 start-page: 1107 year: 2004 ident: 1399_CR24 publication-title: Math. Comp. doi: 10.1090/S0025-5718-03-01583-7 – volume: 8 start-page: 105 year: 1988 ident: 1399_CR44 publication-title: IMA J. Numer. Anal. doi: 10.1093/imanum/8.1.105 – volume: 305 start-page: 121 year: 2005 ident: 1399_CR53 publication-title: J. Math. Anal. Appl. doi: 10.1016/j.jmaa.2004.10.059 – volume: 21 start-page: 89 year: 2012 ident: 1399_CR13 publication-title: Acta Numer doi: 10.1017/S0962492912000037 – ident: 1399_CR42 doi: 10.1007/978-3-540-68093-2 – ident: 1399_CR20 doi: 10.1115/1.3424474 – volume: 87 start-page: 179 year: 2017 ident: 1399_CR17 publication-title: Integr. Equat. Operat. Th. doi: 10.1007/s00020-017-2342-5 – ident: 1399_CR50 doi: 10.1007/978-3-0348-0034-1 – volume: 45 start-page: 38 year: 2003 ident: 1399_CR52 publication-title: IEEE Ant. Propag. Mag. doi: 10.1109/MAP.2003.1189650 – ident: 1399_CR29 doi: 10.1007/s11075-023-01705-8 – ident: 1399_CR51 doi: 10.4171/019 – volume: 474 start-page: 143 year: 2019 ident: 1399_CR54 publication-title: J. Math. Anal. Appl. doi: 10.1016/j.jmaa.2019.01.036 – volume: 10 start-page: 236 year: 1987 ident: 1399_CR48 publication-title: Integr. Equat. Oper. Th. doi: 10.1007/BF01199079 – ident: 1399_CR18 doi: 10.1112/mtk.12155 – volume: 35 start-page: 648 year: 2001 ident: 1399_CR46 publication-title: Aerosol Sci. Tech. doi: 10.1080/02786820117868 – ident: 1399_CR26 doi: 10.1090/gsm/019 – volume: 82 start-page: 423 year: 2015 ident: 1399_CR14 publication-title: Integr. Equat. Oper. Th. doi: 10.1007/s00020-015-2233-6 – ident: 1399_CR11 – volume: 61 start-page: 414 year: 2015 ident: 1399_CR16 publication-title: Mathematika doi: 10.1112/S0025579314000278
SSID	ssj0017641
Score	2.422708
Snippet	Sound-soft fractal screens can scatter acoustic waves even when they have zero surface measure. To solve such scattering problems we make what appears to be...
SourceID	proquest crossref springer
SourceType	Aggregation Database Enrichment Source Index Database Publisher
StartPage	463
SubjectTerms	Acoustic scattering Acoustic waves Acoustics Basis functions Boundary element method Boundary integral method Convergence Estimates Fractals Function space Integral equations Mathematical analysis Mathematical and Computational Engineering Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Numerical Analysis Numerical and Computational Physics Regularity Screens Simulation Theoretical
Title	A Hausdorff-measure boundary element method for acoustic scattering by fractal screens
URI	https://link.springer.com/article/10.1007/s00211-024-01399-7 https://www.proquest.com/docview/3042436816
Volume	156
WOSCitedRecordID	wos001172025600001&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D
hasFullText	1
inHoldings	1
isFullTextHit
isPrint
journalDatabaseRights	– providerCode: PRVAVX databaseName: SpringerLINK Contemporary 1997-Present customDbUrl: eissn: 0945-3245 dateEnd: 99991231 omitProxy: false ssIdentifier: ssj0017641 issn: 0029-599X databaseCode: RSV dateStart: 19970101 isFulltext: true titleUrlDefault: https://link.springer.com/search?facet-content-type=%22Journal%22 providerName: Springer Nature
link	http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwnV3NS8MwFH_o9KAHp1NxOiUHbxpYkzZpj0OUXRzix9itJGkCgm7SbsL-e5P0Yygq6Llpmr685L2XvPf7AZwTkcR22RBsndEIh4IonAglsZbCxDQTwujQk03w0SieTJK7qiisqLPd6ytJv1M3xW7OHNnQl7isCWtWMV-HDWvuYkfYcP8wbu4OOAuDOrEjSpJJVSrzfR-fzdHKx_xyLeqtzU37f-PchZ3Ku0SDUh32YE1PO9CumRtQtZA7sH3boLUW-zAeoKFYFNksNwa_loeGSHq-pXyJdJlgjkquaWSdXGR3UU8Chgrl4Tnt8JBcIuMqruzn7U7kouMDeLq5frwa4opvASvrtsxxFBNpI0RKaBYraax9Y5mQXJjEGM3ifsZ0XzAbr4UqEdQQKiW3f6wY0YRHUtBDaE1nU30ESKu-ppLHmQyZgz-UVISB1swoGnEtki4EtdhTVYGRO06Ml7SBUfZiTK0YUy_GlHfhonnnrYTi-LV1r57NtFqWRerObhzkfsC6cFnP3urxz70d_635CWwRrwBOB3rQmucLfQqb6n3-XORnXl0_ABWZ5M0
linkProvider	Springer Nature
linkToHtml	http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwnV1LS8QwEB50FdSD6xPXZw7eNLBN2qQ9LqKsuC7ii72VJE1A0F1pV8F_b5I-RFFBz03TdDLJzCQz3wdwSEQS22VDsHVGIxwKonAilMRaChPTTAijQ082wYfDeDRKrqqisKLOdq-vJP1O3RS7OXNkQ1_isiasWcV8FuZCa7EcYv71zX1zd8BZGNSJHVGSjKpSme_7-GyOPnzML9ei3tqctf83zhVYrrxL1CvVYRVm9HgN2jVzA6oW8hosXTZorcU63PdQX7wU2SQ3Bj-Vh4ZIer6l_A3pMsEclVzTyDq5yO6ingQMFcrDc9rhIfmGjKu4sp-3O5GLjjfg7uz09qSPK74FrKzbMsVRTKSNECmhWayksfaNZUJyYRJjNIu7GdNdwWy8FqpEUEOolNz-sWJEEx5JQTehNZ6M9RYgrbqaSh5nMmQO_lBSEQZaM6NoxLVIOhDUYk9VBUbuODEe0wZG2YsxtWJMvRhT3oGj5p3nEorj19a79Wym1bIsUnd24yD3A9aB43r2Ph7_3Nv235ofwEL_9nKQDs6HFzuwSLwyOH3YhdY0f9F7MK9epw9Fvu9V9x3Xiuex
linkToPdf	http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwpV1LSwMxEB60iujBt1itmoM3DbbJbrJ7FLVU1FLwQW9Lkk1A0Fa6Vei_N8k-qqKCeN5sNptMMjOZme8DOCQijuy2IdgaoyEOBFE4FkpiLYWJaCqE0YEnm-DdbtTvx70PVfw-270MSeY1DQ6laTA-eUnNSVX45lSTdYOJy6CwKhbzWZgLXCK989dvH6o4AmdBq0zyCOO4X5TNfN_HZ9U0tTe_hEi95mmv_H_Mq7BcWJ3oNBeTNZjRg3VYKRkdULHB12HppkJxzTbg4RR1xGuWDkfG4Of8MhFJz8M0miCdJ56jnIMaWeMX2dPVk4OhTHnYTjtUJCfIuEos-3l7QjmveRPu2xd3Zx1c8DBgZc2ZMQ4jIq3nSAlNIyWN1XssFZILExujWdRMmW4KZv24QMWCGkKl5PaPFSOa8FAKugW1wXCgtwFp1dRU8iiVAXOwiJKKoKU1M4qGXIu4Dq1yCRJVgJQ7roynpIJX9tOY2GlM_DQmvA5H1TsvOUTHr60b5comxXbNEnen46D4W6wOx-VKTh__3NvO35ofwELvvJ1cX3avdmGReFlw4tCA2nj0qvdgXr2NH7PRvpfidw9D8JU
openUrl	ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=A+Hausdorff-measure+boundary+element+method+for+acoustic+scattering+by+fractal+screens&rft.jtitle=Numerische+Mathematik&rft.au=Caetano%2C+A.+M.&rft.au=Chandler-Wilde%2C+S.+N.&rft.au=Gibbs%2C+A.&rft.au=Hewett%2C+D.+P.&rft.date=2024-04-01&rft.issn=0029-599X&rft.eissn=0945-3245&rft.volume=156&rft.issue=2&rft.spage=463&rft.epage=532&rft_id=info:doi/10.1007%2Fs00211-024-01399-7&rft.externalDBID=n%2Fa&rft.externalDocID=10_1007_s00211_024_01399_7
thumbnail_l	http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=0029-599X&client=summon
thumbnail_m	http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=0029-599X&client=summon
thumbnail_s	http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=0029-599X&client=summon