A new augmented immersed finite element method without using SVD interpolations

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• 作者：Haifeng Ji ; Jinru Chen ; Zhilin Li
• 关键词：Interface problem ; Piecewise constant coefficient ; Immersed finite element ; Augmented immersed finite element method ; Poisson equation on irregular domain ; Fast poisson solver ; Least squares interpolation using SVD ; 65N15 ; 65N30 ; 35J60
• 刊名：Numerical Algorithms
• 出版年：2016
• 出版时间：February 2016
• 年：2016
• 卷：71
• 期：2
• 页码：395-416
• 全文大小：2,225 KB
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• 作者单位：Haifeng Ji (1)
  Jinru Chen (1)
  Zhilin Li (1) (2)
  
  1. School of Mathematical Sciences, Jiangsu Key Laboratory for NSLSCS, Nanjing Normal University, Nanjing, 210023, China
  2. Department of Mathematics, North Carolina State University, Raleigh, NC, 27695, USA
  
• 刊物类别：Computer Science
• 刊物主题：Numeric Computing
  Algorithms
  Mathematics
  Algebra
  Theory of Computation
  
• 出版者：Springer U.S.
• ISSN：1572-9265

文摘

Augmented immersed interface methods have been developed recently for interface problems and problems on irregular domains including CFD applications with free boundaries and moving interfaces. In an augmented method, one or several augmented variables are introduced along the interface or boundary so that one can get efficient discretizations. The augmented variables should be chosen such that the interface or boundary conditions are satisfied. The key to the success of the augmented methods often relies on the interpolation scheme to couple the augmented variables with the governing differential equations through the interface or boundary conditions. This has been done using a least squares interpolation (under-determined) for which the singular value decomposition (SVD) is used to solve for the interpolation coefficients. In this paper, based on properties of the finite element method, a new augmented immersed finite element method (IFEM) that does not need the interpolations is proposed for elliptic interface problems that have a piecewise constant coefficient. Thus the new augmented method is more efficient and simple than the old one that uses interpolations. The method then is extended to Poisson equations on irregular domains with a Dirichlet boundary condition. Numerical experiments with arbitrary interfaces/irregular domains and large jump ratios are provided to demonstrate the accuracy and the efficiency of the new augmented methods. Numerical results also show that the number of GMRES iterations is independent of the mesh size and nearly independent of the jump in the coefficient. Keywords Interface problem Piecewise constant coefficient Immersed finite element Augmented immersed finite element method Poisson equation on irregular domain Fast poisson solver Least squares interpolation using SVD