非均匀介质中声波散射的体积积分方程和单迹公式

IF 1 4区数学 Q3 MATHEMATICS, APPLIED

Computational Methods in Applied Mathematics Pub Date : 2023-05-06 DOI:10.1515/cmam-2022-0119

Ignacio Labarca, R. Hiptmair

{"title":"非均匀介质中声波散射的体积积分方程和单迹公式","authors":"Ignacio Labarca, R. Hiptmair","doi":"10.1515/cmam-2022-0119","DOIUrl":null,"url":null,"abstract":"Abstract We study frequency domain acoustic scattering at a bounded, penetrable, and inhomogeneous obstacle Ω − ⊂ R d \\Omega^{-}\\subset\\mathbb{R}^{d} , d = 2 , 3 d=2,3 . By defining constant reference coefficients, a representation formula for the pressure field is derived. It contains a volume integral operator, related to the one in the Lippmann–Schwinger equation. Besides, it features integral operators defined on ∂ Ω − \\partial\\Omega^{-} and closely related to boundary integral equations of single-trace formulations (STF) for transmission problems with piecewise constant coefficients. We show well-posedness of the continuous variational formulation and asymptotic convergence of Galerkin discretizations. Numerical experiments in 2D validate our expected convergence rates.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":1.0000,"publicationDate":"2023-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Volume Integral Equations and Single-Trace Formulations for Acoustic Wave Scattering in an Inhomogeneous Medium\",\"authors\":\"Ignacio Labarca, R. Hiptmair\",\"doi\":\"10.1515/cmam-2022-0119\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Abstract We study frequency domain acoustic scattering at a bounded, penetrable, and inhomogeneous obstacle Ω − ⊂ R d \\\\Omega^{-}\\\\subset\\\\mathbb{R}^{d} , d = 2 , 3 d=2,3 . By defining constant reference coefficients, a representation formula for the pressure field is derived. It contains a volume integral operator, related to the one in the Lippmann–Schwinger equation. Besides, it features integral operators defined on ∂ Ω − \\\\partial\\\\Omega^{-} and closely related to boundary integral equations of single-trace formulations (STF) for transmission problems with piecewise constant coefficients. We show well-posedness of the continuous variational formulation and asymptotic convergence of Galerkin discretizations. Numerical experiments in 2D validate our expected convergence rates.\",\"PeriodicalId\":48751,\"journal\":{\"name\":\"Computational Methods in Applied Mathematics\",\"volume\":\" \",\"pages\":\"\"},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2023-05-06\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Computational Methods in Applied Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1515/cmam-2022-0119\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Computational Methods in Applied Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/cmam-2022-0119","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}

引用次数: 0

摘要

摘要我们研究了有界、可穿透和非均匀障碍物Ω−⊂Rd\Omega^｛-｝\subet \mathbb｛R｝^｛d｝，d=2，3d=2，3的频域声散射。通过定义常数参考系数，导出了压力场的表示公式。它包含一个体积积分算子，与Lippmann–Schwinger方程中的算子有关。此外，它还具有定义在¦ΒΩ−\partial\Omega^｛-｝上的积分算子，并与分段常系数传输问题的单迹公式（STF）的边界积分方程密切相关。我们证明了连续变分公式的适定性和Galerkin离散化的渐近收敛性。二维数值实验验证了我们的预期收敛速度。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文本刊更多论文

Volume Integral Equations and Single-Trace Formulations for Acoustic Wave Scattering in an Inhomogeneous Medium

Abstract We study frequency domain acoustic scattering at a bounded, penetrable, and inhomogeneous obstacle Ω − ⊂ R d \Omega^{-}\subset\mathbb{R}^{d} , d = 2 , 3 d=2,3 . By defining constant reference coefficients, a representation formula for the pressure field is derived. It contains a volume integral operator, related to the one in the Lippmann–Schwinger equation. Besides, it features integral operators defined on ∂ Ω − \partial\Omega^{-} and closely related to boundary integral equations of single-trace formulations (STF) for transmission problems with piecewise constant coefficients. We show well-posedness of the continuous variational formulation and asymptotic convergence of Galerkin discretizations. Numerical experiments in 2D validate our expected convergence rates.

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助

来源期刊

Computational Methods in Applied Mathematics MATHEMATICS, APPLIED-

CiteScore

2.40

自引率

7.70%

发文量

期刊介绍： The highly selective international mathematical journal Computational Methods in Applied Mathematics　(CMAM) considers original mathematical contributions to computational methods and numerical analysis with applications mainly related to PDEs. CMAM seeks to be interdisciplinary while retaining the common thread of numerical analysis, it is intended to be readily readable and meant for a wide circle of researchers in applied mathematics. The journal is published by De Gruyter on behalf of the Institute of Mathematics of the National Academy of Science of Belarus.