{"title":"A new space transformed finite element method for elliptic interface problems in Rn","authors":"","doi":"10.1016/j.cam.2024.116277","DOIUrl":null,"url":null,"abstract":"<div><p>Interface problems, where distinct materials or physical domains meet, pose significant challenges in numerical simulations due to the discontinuities and sharp gradients across interfaces. Traditional finite element methods struggle to capture such behavior accurately. A new space transformed finite element method (ST-FEM) is developed for solving elliptic interface problems in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>. A homeomorphic stretching transformation is introduced to obtain an equivalent problem in the transformed domain which can be solved easily, and the solution can be projected back to original domain by the inverse transformation. Compared with the existing methods, this new scheme has capability of handling discontinuities across the interface. The proposed approach has advantages in circumventing interface approximation properties and reducing the degree of freedom. We initially develop ST-FEM for elliptic problems and subsequently expand upon this concept to address elliptic interface problems. We prove optimal a priori error estimates in the <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> and <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> norms, and quasi-optimal error estimate for the maximum norm. Finally, numerical experiments demonstrate the superior accuracy and convergence properties of the ST-FEM when compared to the standard finite element method. The interface is assumed to be a <span><math><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></math></span>-sphere, nevertheless, our analysis can cover symmetric domains such as an ellipsoid or a cylinder.</p></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":2.1000,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Computational and Applied Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0377042724005260","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
Interface problems, where distinct materials or physical domains meet, pose significant challenges in numerical simulations due to the discontinuities and sharp gradients across interfaces. Traditional finite element methods struggle to capture such behavior accurately. A new space transformed finite element method (ST-FEM) is developed for solving elliptic interface problems in . A homeomorphic stretching transformation is introduced to obtain an equivalent problem in the transformed domain which can be solved easily, and the solution can be projected back to original domain by the inverse transformation. Compared with the existing methods, this new scheme has capability of handling discontinuities across the interface. The proposed approach has advantages in circumventing interface approximation properties and reducing the degree of freedom. We initially develop ST-FEM for elliptic problems and subsequently expand upon this concept to address elliptic interface problems. We prove optimal a priori error estimates in the and norms, and quasi-optimal error estimate for the maximum norm. Finally, numerical experiments demonstrate the superior accuracy and convergence properties of the ST-FEM when compared to the standard finite element method. The interface is assumed to be a -sphere, nevertheless, our analysis can cover symmetric domains such as an ellipsoid or a cylinder.
期刊介绍:
The Journal of Computational and Applied Mathematics publishes original papers of high scientific value in all areas of computational and applied mathematics. The main interest of the Journal is in papers that describe and analyze new computational techniques for solving scientific or engineering problems. Also the improved analysis, including the effectiveness and applicability, of existing methods and algorithms is of importance. The computational efficiency (e.g. the convergence, stability, accuracy, ...) should be proved and illustrated by nontrivial numerical examples. Papers describing only variants of existing methods, without adding significant new computational properties are not of interest.
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