TY - JOUR
T1 - Analysis of a Fully Discrete Finite Element Method for Parabolic Interface Problems with Nonsmooth Initial Data
AU - Wang , Kai
AU - Wang , Na
JO - Journal of Computational Mathematics
VL - 5
SP - 777
EP - 793
PY - 2022
DA - 2022/05
SN - 40
DO - http://doi.org/10.4208/jcm.2101-m2020-0075
UR - https://global-sci.org/intro/article_detail/jcm/20547.html
KW - Parabolic interface problem, Finite element method, Backward difference formulae, Error estimate, Nonsmooth initial data.
AB - <p style="text-align: justify;">This article concerns numerical approximation of a parabolic interface problem with general $L^2$ initial value. The problem is discretized by a finite element method with a quasi-uniform triangulation of the domain fitting the interface, with piecewise linear approximation to the interface. The semi-discrete finite element problem is furthermore discretized in time by the $k$-step backward difference formula&nbsp; with $ k=1,\ldots,6 $. To maintain high-order convergence in time for possibly nonsmooth $L^2$ initial value, we modify the standard backward difference formula at the first $k-1$ time levels by using a method recently developed for fractional evolution equations. An error bound of $\mathcal{O}(t_n^{-k}\tau^k+t_n^{-1}h^2|\log h|)$ is established for the fully discrete finite element method for general $L^2$ initial data.</p>