{"title":"奇异扰动反应扩散问题二元有限元法的后验误差估计","authors":"JaEun Ku, Martin Stynes","doi":"10.1007/s10543-024-01008-x","DOIUrl":null,"url":null,"abstract":"<p><i>A posteriori</i> error estimates are established for a two-step dual finite element method for singularly perturbed reaction–diffusion problems. The method can be considered as a modified least-squares finite element method. The least-squares functional is the basis for our residual-type a posteriori error estimators, which are shown to be reliable and efficient with respect to the error in an energy-type norm. Moreover, guaranteed upper bounds for the errors in the computed primary and dual variables are derived; these bounds are then used to drive an adaptive algorithm for our finite element method, yielding any desired accuracy. Our theory does not require the meshes generated to be shape-regular. Numerical experiments show the effectiveness of our a posteriori estimators.</p>","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A posteriori error estimates for a dual finite element method for singularly perturbed reaction–diffusion problems\",\"authors\":\"JaEun Ku, Martin Stynes\",\"doi\":\"10.1007/s10543-024-01008-x\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p><i>A posteriori</i> error estimates are established for a two-step dual finite element method for singularly perturbed reaction–diffusion problems. The method can be considered as a modified least-squares finite element method. The least-squares functional is the basis for our residual-type a posteriori error estimators, which are shown to be reliable and efficient with respect to the error in an energy-type norm. Moreover, guaranteed upper bounds for the errors in the computed primary and dual variables are derived; these bounds are then used to drive an adaptive algorithm for our finite element method, yielding any desired accuracy. Our theory does not require the meshes generated to be shape-regular. Numerical experiments show the effectiveness of our a posteriori estimators.</p>\",\"PeriodicalId\":1,\"journal\":{\"name\":\"Accounts of Chemical Research\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":16.4000,\"publicationDate\":\"2024-02-05\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Accounts of Chemical Research\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s10543-024-01008-x\",\"RegionNum\":1,\"RegionCategory\":\"化学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"CHEMISTRY, MULTIDISCIPLINARY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10543-024-01008-x","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
A posteriori error estimates for a dual finite element method for singularly perturbed reaction–diffusion problems
A posteriori error estimates are established for a two-step dual finite element method for singularly perturbed reaction–diffusion problems. The method can be considered as a modified least-squares finite element method. The least-squares functional is the basis for our residual-type a posteriori error estimators, which are shown to be reliable and efficient with respect to the error in an energy-type norm. Moreover, guaranteed upper bounds for the errors in the computed primary and dual variables are derived; these bounds are then used to drive an adaptive algorithm for our finite element method, yielding any desired accuracy. Our theory does not require the meshes generated to be shape-regular. Numerical experiments show the effectiveness of our a posteriori estimators.
期刊介绍：
Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance.
Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.
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