{"title":"An hp-version of the discontinuous Galerkin method for fractional integro-differential equations with weakly singular kernels","authors":"Yanping Chen, Zhenrong Chen, Yunqing Huang","doi":"10.1007/s10543-024-01026-9","DOIUrl":null,"url":null,"abstract":"<p>This paper suggests an <i>hp</i>-discontinuous Galerkin approach for the fractional integro-differential equations with weakly singular kernels. The key idea behind our method is to first convert the fractional integro-differential equations into the second kind of Volterra integral equations, and then solve the equivalent integral equations using the <i>hp</i>-discontinuous Galerkin method. We establish prior error bounds in the <span>\\(L^{2}\\)</span>-norm that is entirely explicit about the local mesh sizes, local polynomial degrees, and local regularities of the exact solutions. The use of geometrically refined meshes and linearly increasing approximation orders demonstrates, in particular, that exponential convergence is achievable for solutions with endpoint singularities. Numerical results indicate the usefulness of the proposed method.</p>","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-07-04","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/s10543-024-01026-9","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0
Abstract
This paper suggests an hp-discontinuous Galerkin approach for the fractional integro-differential equations with weakly singular kernels. The key idea behind our method is to first convert the fractional integro-differential equations into the second kind of Volterra integral equations, and then solve the equivalent integral equations using the hp-discontinuous Galerkin method. We establish prior error bounds in the \(L^{2}\)-norm that is entirely explicit about the local mesh sizes, local polynomial degrees, and local regularities of the exact solutions. The use of geometrically refined meshes and linearly increasing approximation orders demonstrates, in particular, that exponential convergence is achievable for solutions with endpoint singularities. Numerical results indicate the usefulness of the proposed method.
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.