A Well-Conditioned, Nonconforming Nitsche's Extended Finite Element Method for Elliptic Interface Problems
Year: 2020
Author: Fei Song, Xiaoxiao He, Weibing Deng, Fei Song, Weibing Deng
Numerical Mathematics: Theory, Methods and Applications, Vol. 13 (2020), Iss. 1 : pp. 99–130
Abstract
In this paper, we introduce a nonconforming Nitsche's extended finite element method (NXFEM) for elliptic interface problems on unfitted triangulation elements. The solution on each side of the interface is separately expanded in the standard nonconforming piecewise linear polynomials with the edge averages as degrees of freedom. The jump conditions on the interface and the discontinuities on the cut edges (the segment of edges cut by the interface) are weakly enforced by the Nitsche's approach. In the method, the harmonic weighted fluxes are used and the extra stabilization terms on the interface edges and cut edges are added to guarantee the stability and the well conditioning. We prove that the convergence order of the errors in energy and $L^2$ norms are optimal. Moreover, the errors are independent of the position of the interface relative to the mesh and the ratio of the discontinuous coefficients. Furthermore, we prove that the condition number of the system matrix is independent of the interface position. Numerical examples are given to confirm the theoretical results.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/nmtma.OA-2019-0053
Numerical Mathematics: Theory, Methods and Applications, Vol. 13 (2020), Iss. 1 : pp. 99–130
Published online: 2020-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 32
Keywords: Elliptic interface problems NXFEM nonconforming finite element condition number.