Volume 13, Issue 1
A Well-Conditioned, Nonconforming Nitsche's Extended Finite Element Method for Elliptic Interface Problems

Numer. Math. Theor. Meth. Appl., 13 (2020), pp. 99-130.

Published online: 2019-12

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• 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.

• Keywords

Elliptic interface problems, NXFEM, nonconforming finite element, condition number.

65N12, 65N15, 65N30

hxxmath@163.com (Xiaoxiao He)

songfei@njfu.edu.cn (Fei Song)

wbdeng@nju.edu.cn (Weibing Deng)

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@Article{NMTMA-13-99, author = {He , Xiaoxiao and Song , Fei and Deng , Weibing}, title = {A Well-Conditioned, Nonconforming Nitsche's Extended Finite Element Method for Elliptic Interface Problems}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2019}, volume = {13}, number = {1}, pages = {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.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2019-0053}, url = {http://global-sci.org/intro/article_detail/nmtma/13433.html} }
TY - JOUR T1 - A Well-Conditioned, Nonconforming Nitsche's Extended Finite Element Method for Elliptic Interface Problems AU - He , Xiaoxiao AU - Song , Fei AU - Deng , Weibing JO - Numerical Mathematics: Theory, Methods and Applications VL - 1 SP - 99 EP - 130 PY - 2019 DA - 2019/12 SN - 13 DO - http://doi.org/10.4208/nmtma.OA-2019-0053 UR - https://global-sci.org/intro/article_detail/nmtma/13433.html KW - Elliptic interface problems, NXFEM, nonconforming finite element, condition number. AB -

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.

Xiaoxiao He, Fei Song & Weibing Deng. (2019). A Well-Conditioned, Nonconforming Nitsche's Extended Finite Element Method for Elliptic Interface Problems. Numerical Mathematics: Theory, Methods and Applications. 13 (1). 99-130. doi:10.4208/nmtma.OA-2019-0053
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