@Article{cmes.2020.09831,
AUTHOR = {Pengfei Zhu, Qinghui Zhang},
TITLE = {BDF Schemes in Stable Generalized Finite Element Methods for Parabolic Interface Problems with Moving Interfaces},
JOURNAL = {Computer Modeling in Engineering \& Sciences},
VOLUME = {124},
YEAR = {2020},
NUMBER = {1},
PAGES = {107--127},
URL = {http://www.techscience.com/CMES/v124n1/39384},
ISSN = {1526-1506},
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.},
DOI = {10.32604/cmes.2020.09831}
}