Volume 40, Issue 5
Analysis of a Fully Discrete Finite Element Method for Parabolic Interface Problems with Nonsmooth Initial Data

J. Comp. Math., 40 (2022), pp. 781-797.

Published online: 2022-05

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• Abstract

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  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.

• Keywords

Parabolic interface problem, Finite element method, Backward difference formulae, Error estimate, Nonsmooth initial data.

65M60, 65N30, 65N15

wangk33@sustech.edu.cn (Kai Wang)

wangna@csrc.ac.cn (Na Wang)

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@Article{JCM-40-781, author = {Kai and Wang and wangk33@sustech.edu.cn and 23359 and Department of Mathematics, Southern University of Science and Technology, Shenzhen 518005, China and Kai Wang and Na and Wang and wangna@csrc.ac.cn and 23360 and Applied and Computational Mathematics Division, Beijing Computational Science Research Center, Beijing 100193, China and Na Wang}, title = {Analysis of a Fully Discrete Finite Element Method for Parabolic Interface Problems with Nonsmooth Initial Data}, journal = {Journal of Computational Mathematics}, year = {2022}, volume = {40}, number = {5}, pages = {781--797}, abstract = {

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  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.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2101-m2020-0075}, url = {http://global-sci.org/intro/article_detail/jcm/20547.html} }
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 - 781 EP - 797 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 -

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  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.

Kai Wang & Na Wang. (2022). Analysis of a Fully Discrete Finite Element Method for Parabolic Interface Problems with Nonsmooth Initial Data. Journal of Computational Mathematics. 40 (5). 781-797. doi:10.4208/jcm.2101-m2020-0075
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