{"title":"求解多谐问题的多层次基本解法","authors":"Andreas Karageorghis , C.S. Chen","doi":"10.1016/j.cam.2024.116220","DOIUrl":null,"url":null,"abstract":"<div><p>We consider a multi–level method of fundamental solutions for solving polyharmonic problems governed by <span><math><mrow><msup><mrow><mi>Δ</mi></mrow><mrow><mi>N</mi></mrow></msup><mi>u</mi><mo>=</mo><mn>0</mn><mo>,</mo><mspace></mspace><mi>N</mi><mo>∈</mo><mi>N</mi><mo>∖</mo><mrow><mo>{</mo><mn>1</mn><mo>}</mo></mrow></mrow></math></span> in both two and three dimensions. Instead of approximating the solution with linear combinations of <span><math><mi>N</mi></math></span> fundamental solutions, we show that, with appropriate deployments of the source points, it is possible to employ an approximation involving only the fundamental solution of the operator <span><math><msup><mrow><mi>Δ</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span>. To determine the optimal position of the source points, we apply the recently developed effective condition number method. In addition, we show that when the proposed technique is applied to boundary value problems in circular or axisymmetric domains, with appropriate distributions of boundary and source points, it lends itself to the application of matrix decomposition algorithms. The results of several numerical tests are presented and analysed.</p></div>","PeriodicalId":2,"journal":{"name":"ACS Applied Bio Materials","volume":null,"pages":null},"PeriodicalIF":4.6000,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Multi–level method of fundamental solutions for solving polyharmonic problems\",\"authors\":\"Andreas Karageorghis , C.S. Chen\",\"doi\":\"10.1016/j.cam.2024.116220\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We consider a multi–level method of fundamental solutions for solving polyharmonic problems governed by <span><math><mrow><msup><mrow><mi>Δ</mi></mrow><mrow><mi>N</mi></mrow></msup><mi>u</mi><mo>=</mo><mn>0</mn><mo>,</mo><mspace></mspace><mi>N</mi><mo>∈</mo><mi>N</mi><mo>∖</mo><mrow><mo>{</mo><mn>1</mn><mo>}</mo></mrow></mrow></math></span> in both two and three dimensions. Instead of approximating the solution with linear combinations of <span><math><mi>N</mi></math></span> fundamental solutions, we show that, with appropriate deployments of the source points, it is possible to employ an approximation involving only the fundamental solution of the operator <span><math><msup><mrow><mi>Δ</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span>. To determine the optimal position of the source points, we apply the recently developed effective condition number method. In addition, we show that when the proposed technique is applied to boundary value problems in circular or axisymmetric domains, with appropriate distributions of boundary and source points, it lends itself to the application of matrix decomposition algorithms. The results of several numerical tests are presented and analysed.</p></div>\",\"PeriodicalId\":2,\"journal\":{\"name\":\"ACS Applied Bio Materials\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":4.6000,\"publicationDate\":\"2024-08-22\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"ACS Applied Bio Materials\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0377042724004692\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATERIALS SCIENCE, BIOMATERIALS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACS Applied Bio Materials","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0377042724004692","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATERIALS SCIENCE, BIOMATERIALS","Score":null,"Total":0}
引用次数: 0
摘要
我们考虑了一种多层次基本解法,用于解决二维和三维中受 ΔNu=0,N∈N∖{1} 控制的多谐问题。我们不再用 N 个基本解的线性组合来近似求解,而是表明,在适当部署源点的情况下,可以采用只涉及算子 ΔN 基本解的近似方法。为了确定源点的最佳位置,我们采用了最近开发的有效条件数法。此外,我们还表明,当将所提出的技术应用于圆形或轴对称域中的边界值问题时,如果边界点和源点分布适当,则可以应用矩阵分解算法。我们介绍并分析了几个数值测试的结果。
Multi–level method of fundamental solutions for solving polyharmonic problems
We consider a multi–level method of fundamental solutions for solving polyharmonic problems governed by in both two and three dimensions. Instead of approximating the solution with linear combinations of fundamental solutions, we show that, with appropriate deployments of the source points, it is possible to employ an approximation involving only the fundamental solution of the operator . To determine the optimal position of the source points, we apply the recently developed effective condition number method. In addition, we show that when the proposed technique is applied to boundary value problems in circular or axisymmetric domains, with appropriate distributions of boundary and source points, it lends itself to the application of matrix decomposition algorithms. The results of several numerical tests are presented and analysed.