Open Access
ARTICLE
BDF Schemes in Stable Generalized Finite Element Methods for Parabolic Interface Problems with Moving Interfaces
Pengfei Zhu1, Qinghui Zhang2,*
1 School of Data and Computer Science, Sun Yat-Sen University, Guangzhou, 510006, China
2 Guangdong Province Key Laboratory of Computational Science and School of Data and Computer Science, Sun Yat-Sen University, Guangzhou, 510006, China
* Corresponding Author: Qinghui Zhang. Email:
Computer Modeling in Engineering & Sciences 2020, 124(1), 107-127. https://doi.org/10.32604/cmes.2020.09831
Received 21 January 2020; Accepted 23 March 2020; Issue published 19 June 2020
Abstract
There are several difficulties in generalized/extended finite element
methods (GFEM/XFEM) for moving interface problems. First, the GFEM/XFEM
may be unstable in a sense that condition numbers of system matrices could be
much bigger than those of standard FEM. Second, they may not be robust in that
the condition numbers increase rapidly as interface curves approach edges of
meshes. Furthermore, time stepping schemes need carrying out carefully since
both enrichment functions and enriched nodes in the GFEM/XFEM vary in time.
This paper is devoted to proposing the stable and robust GFEM/XFEM with effi-
cient time stepping schemes for the parabolic interface problems with moving
interfaces. A so-called stable GFEM (SGFEM) developed for elliptical interface
problems is extended to the parabolic interface problems for spatial discretizations; while backward difference formulae (BDF) are used for the time stepping.
Numerical studies demonstrate that the SGFEM with the first and second order
BDF (also known as backward Euler method and BDF2) is stable, robust, and
achieves optimal convergence rates. Comparisons of the proposed SGFEM with
various commonly-used GFEM/XFEM are made, which show advantages of
the SGFEM over the other GFEM/XFEM in aspects of stability, robustness,
and convergence.
Keywords
Cite This Article
Zhu, P., Zhang, Q. (2020). BDF Schemes in Stable Generalized Finite Element Methods for Parabolic Interface Problems with Moving Interfaces.
CMES-Computer Modeling in Engineering & Sciences, 124(1), 107–127. https://doi.org/10.32604/cmes.2020.09831