Xiaoxiao He, Yanping Chen, Haifeng Ji, Haijin Wang
{"title":"Superconvergence of unfitted Rannacher-Turek nonconforming element for elliptic interface problems","authors":"Xiaoxiao He, Yanping Chen, Haifeng Ji, Haijin Wang","doi":"10.1016/j.apnum.2024.05.016","DOIUrl":null,"url":null,"abstract":"<div><p>The main aim of this paper is to study the superconvergence of nonconforming Rannacher-Turek finite element for elliptic interface problems under unfitted square meshes. In particular, we analyze its superclose property between the gradient of the numerical solution and the gradient of the interpolation of the exact solution. Moreover, we introduce a postprocessing interpolation operator which is applied to numerical solution, and we prove that the postprocessed gradient converges to the exact gradient with a superconvergent rate <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>h</mi></mrow><mrow><mfrac><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mo>)</mo></math></span>. Finally, numerical results coincide with our theoretical analysis, and they show that the error estimates do not depend on the ratio of the discontinuous coefficients.</p></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"203 ","pages":"Pages 32-51"},"PeriodicalIF":2.4000,"publicationDate":"2024-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Applied Numerical Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0168927424001223","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2024/5/23 0:00:00","PubModel":"Epub","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
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Abstract
The main aim of this paper is to study the superconvergence of nonconforming Rannacher-Turek finite element for elliptic interface problems under unfitted square meshes. In particular, we analyze its superclose property between the gradient of the numerical solution and the gradient of the interpolation of the exact solution. Moreover, we introduce a postprocessing interpolation operator which is applied to numerical solution, and we prove that the postprocessed gradient converges to the exact gradient with a superconvergent rate . Finally, numerical results coincide with our theoretical analysis, and they show that the error estimates do not depend on the ratio of the discontinuous coefficients.
期刊介绍：
The purpose of the journal is to provide a forum for the publication of high quality research and tutorial papers in computational mathematics. In addition to the traditional issues and problems in numerical analysis, the journal also publishes papers describing relevant applications in such fields as physics, fluid dynamics, engineering and other branches of applied science with a computational mathematics component. The journal strives to be flexible in the type of papers it publishes and their format. Equally desirable are:
(i) Full papers, which should be complete and relatively self-contained original contributions with an introduction that can be understood by the broad computational mathematics community. Both rigorous and heuristic styles are acceptable. Of particular interest are papers about new areas of research, in which other than strictly mathematical arguments may be important in establishing a basis for further developments.
(ii) Tutorial review papers, covering some of the important issues in Numerical Mathematics, Scientific Computing and their Applications. The journal will occasionally publish contributions which are larger than the usual format for regular papers.
(iii) Short notes, which present specific new results and techniques in a brief communication.
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