Charles L. Epstein, Leslie Greengard, Jeremy Hoskins, Shidong Jiang, Manas Rachh
{"title":"扰动半空间中带有狄里赫特边界条件的亥姆霍兹方程的坐标复合化","authors":"Charles L. Epstein, Leslie Greengard, Jeremy Hoskins, Shidong Jiang, Manas Rachh","doi":"arxiv-2409.06988","DOIUrl":null,"url":null,"abstract":"We present a new complexification scheme based on the classical double layer\npotential for the solution of the Helmholtz equation with Dirichlet boundary\nconditions in compactly perturbed half-spaces in two and three dimensions. The\nkernel for the double layer potential is the normal derivative of the\nfree-space Green's function, which has a well-known analytic continuation into\nthe complex plane as a function of both target and source locations. Here, we\nprove that - when the incident data are analytic and satisfy a precise\nasymptotic estimate - the solution to the boundary integral equation itself\nadmits an analytic continuation into specific regions of the complex plane, and\nsatisfies a related asymptotic estimate (this class of data includes both plane\nwaves and the field induced by point sources). We then show that, with a\ncarefully chosen contour deformation, the oscillatory integrals are converted\nto exponentially decaying integrals, effectively reducing the infinite domain\nto a domain of finite size. Our scheme is different from existing methods that\nuse complex coordinate transformations, such as perfectly matched layers, or\nabsorbing regions, such as the gradual complexification of the governing\nwavenumber. More precisely, in our method, we are still solving a boundary\nintegral equation, albeit on a truncated, complexified version of the original\nboundary. In other words, no volumetric/domain modifications are introduced.\nThe scheme can be extended to other boundary conditions, to open wave guides\nand to layered media. We illustrate the performance of the scheme with two and\nthree dimensional examples.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"118 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Coordinate complexification for the Helmholtz equation with Dirichlet boundary conditions in a perturbed half-space\",\"authors\":\"Charles L. Epstein, Leslie Greengard, Jeremy Hoskins, Shidong Jiang, Manas Rachh\",\"doi\":\"arxiv-2409.06988\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We present a new complexification scheme based on the classical double layer\\npotential for the solution of the Helmholtz equation with Dirichlet boundary\\nconditions in compactly perturbed half-spaces in two and three dimensions. The\\nkernel for the double layer potential is the normal derivative of the\\nfree-space Green's function, which has a well-known analytic continuation into\\nthe complex plane as a function of both target and source locations. Here, we\\nprove that - when the incident data are analytic and satisfy a precise\\nasymptotic estimate - the solution to the boundary integral equation itself\\nadmits an analytic continuation into specific regions of the complex plane, and\\nsatisfies a related asymptotic estimate (this class of data includes both plane\\nwaves and the field induced by point sources). We then show that, with a\\ncarefully chosen contour deformation, the oscillatory integrals are converted\\nto exponentially decaying integrals, effectively reducing the infinite domain\\nto a domain of finite size. Our scheme is different from existing methods that\\nuse complex coordinate transformations, such as perfectly matched layers, or\\nabsorbing regions, such as the gradual complexification of the governing\\nwavenumber. More precisely, in our method, we are still solving a boundary\\nintegral equation, albeit on a truncated, complexified version of the original\\nboundary. In other words, no volumetric/domain modifications are introduced.\\nThe scheme can be extended to other boundary conditions, to open wave guides\\nand to layered media. We illustrate the performance of the scheme with two and\\nthree dimensional examples.\",\"PeriodicalId\":501162,\"journal\":{\"name\":\"arXiv - MATH - Numerical Analysis\",\"volume\":\"118 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-11\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Numerical Analysis\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2409.06988\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Numerical Analysis","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.06988","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Coordinate complexification for the Helmholtz equation with Dirichlet boundary conditions in a perturbed half-space
We present a new complexification scheme based on the classical double layer
potential for the solution of the Helmholtz equation with Dirichlet boundary
conditions in compactly perturbed half-spaces in two and three dimensions. The
kernel for the double layer potential is the normal derivative of the
free-space Green's function, which has a well-known analytic continuation into
the complex plane as a function of both target and source locations. Here, we
prove that - when the incident data are analytic and satisfy a precise
asymptotic estimate - the solution to the boundary integral equation itself
admits an analytic continuation into specific regions of the complex plane, and
satisfies a related asymptotic estimate (this class of data includes both plane
waves and the field induced by point sources). We then show that, with a
carefully chosen contour deformation, the oscillatory integrals are converted
to exponentially decaying integrals, effectively reducing the infinite domain
to a domain of finite size. Our scheme is different from existing methods that
use complex coordinate transformations, such as perfectly matched layers, or
absorbing regions, such as the gradual complexification of the governing
wavenumber. More precisely, in our method, we are still solving a boundary
integral equation, albeit on a truncated, complexified version of the original
boundary. In other words, no volumetric/domain modifications are introduced.
The scheme can be extended to other boundary conditions, to open wave guides
and to layered media. We illustrate the performance of the scheme with two and
three dimensional examples.