{"title":"A New Immersed Finite Element Method for Two-Phase Stokes Problems Having Discontinuous Pressure","authors":"Gwanghyun Jo, D. Kwak","doi":"10.1515/cmam-2022-0122","DOIUrl":null,"url":null,"abstract":"Abstract In this paper, we develop a new immersed finite element method (IFEM) for two-phase incompressible Stokes flows. We allow the interface to cut the finite elements. On the noninterface element, the standard Crouzeix–Raviart element and the P 0 {P_{0}} element pair is used. On the interface element, the basis functions developed for scalar interface problems (Kwak et al., An analysis of a broken P 1 {P_{1}} -nonconforming finite element method for interface problems, SIAM J. Numer. Anal. (2010)) are modified in such a way that the coupling between the velocity and pressure variable is different. There are two kinds of basis functions. The first kind of basis satisfies the Laplace–Young condition under the assumption of the continuity of the pressure variable. In the second kind, the velocity is of bubble type and is coupled with the discontinuous pressure, still satisfying the Laplace–Young condition. We remark that in the second kind the pressure variable has two degrees of freedom on each interface element. Therefore, our methods can handle the discontinuous pressure case. Numerical results including the case of the discontinuous pressure variable are provided. We see optimal convergence orders for all examples.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2023-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Computational Methods in Applied Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/cmam-2022-0122","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
Abstract In this paper, we develop a new immersed finite element method (IFEM) for two-phase incompressible Stokes flows. We allow the interface to cut the finite elements. On the noninterface element, the standard Crouzeix–Raviart element and the P 0 {P_{0}} element pair is used. On the interface element, the basis functions developed for scalar interface problems (Kwak et al., An analysis of a broken P 1 {P_{1}} -nonconforming finite element method for interface problems, SIAM J. Numer. Anal. (2010)) are modified in such a way that the coupling between the velocity and pressure variable is different. There are two kinds of basis functions. The first kind of basis satisfies the Laplace–Young condition under the assumption of the continuity of the pressure variable. In the second kind, the velocity is of bubble type and is coupled with the discontinuous pressure, still satisfying the Laplace–Young condition. We remark that in the second kind the pressure variable has two degrees of freedom on each interface element. Therefore, our methods can handle the discontinuous pressure case. Numerical results including the case of the discontinuous pressure variable are provided. We see optimal convergence orders for all examples.
摘要本文提出了一种新的求解两相不可压缩Stokes流的浸入式有限元方法。我们允许界面切割有限元。在非接口元素上,使用标准的Crouzeix–Raviart元素和P0{P_{0}}元素对。在界面单元上,对为标量界面问题开发的基函数(Kwak et al.,An analysis of a breaked P1{P_{1}}-conformant finite element method for interface problems,SIAM J.Numer.Anal.(2010))进行了修改,使得速度和压力变量之间的耦合不同。基函数有两种。在压力变量连续性的假设下,第一类基满足拉普拉斯-杨条件。在第二种情况下,速度是气泡型的,并与不连续压力耦合,仍然满足拉普拉斯-杨条件。我们注意到,在第二类中,压力变量在每个界面元件上有两个自由度。因此,我们的方法可以处理不连续压力的情况。给出了包括不连续压力变量情况下的数值结果。我们看到所有例子的最优收敛阶。
期刊介绍:
The highly selective international mathematical journal Computational Methods in Applied Mathematics (CMAM) considers original mathematical contributions to computational methods and numerical analysis with applications mainly related to PDEs.
CMAM seeks to be interdisciplinary while retaining the common thread of numerical analysis, it is intended to be readily readable and meant for a wide circle of researchers in applied mathematics.
The journal is published by De Gruyter on behalf of the Institute of Mathematics of the National Academy of Science of Belarus.