Optimal preconditioners for Nitsche-XFEM discretizations of interface problems

详细信息    查看全文

• 作者：Christoph Lehrenfeld ; Arnold Reusken
• 关键词：Mathematics Subject Classification65N22 ; 65N30
• 刊名：Numerische Mathematik
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：135
• 期：2
• 页码：313-332
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Numerical Analysis; Mathematics, general; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics, Simulation; Appl.Mathematics/Comput
• 出版者：Springer Berlin Heidelberg
• ISSN：0945-3245
• 卷排序：135

文摘

In the past decade, a combination of unfitted finite elements (or XFEM) with the Nitsche method has become a popular discretization method for elliptic interface problems. This development started with the introduction and analysis of this Nitsche-XFEM technique in the paper (Hansbo and Hansbo, Comput Methods Appl Mech Eng 191:5537–5552, 2002). In general, the resulting linear systems have very large condition numbers, which depend not only on the mesh size h, but also on how the interface intersects the mesh. This paper is concerned with the design and analysis of optimal preconditioners for such linear systems. We propose an additive subspace preconditioner which is optimal in the sense that the resulting condition number is independent of the mesh size h and the interface position. We further show that already the simple diagonal scaling of the stiffness matrix results in a condition number that is bounded by \(ch^{-2}\), with a constant c that does not depend on the location of the interface. Both results are proven for the two-dimensional case. Results of numerical experiments in two and three dimensions are presented, which illustrate the quality of the preconditioner.