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

Kai Wang & Na Wang

DOI: 10.4208/jcm.2101-m2020-0075

J. Comp. Math., 40 (2022), pp. 777-793.

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.

  • AMS Subject Headings

65M60, 65N30, 65N15

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

wangk33@sustech.edu.cn (Kai Wang)

wangna@csrc.ac.cn (Na Wang)

  • BibTex
  • RIS
  • TXT
@Article{JCM-40-777, author = {Wang , Kai and Wang , Na}, 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 = {777--793}, 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 - 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 -

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.

Wang , Kai and Wang , Na. (2022). Analysis of a Fully Discrete Finite Element Method for Parabolic Interface Problems with Nonsmooth Initial Data. Journal of Computational Mathematics. 40 (5). 777-793. doi:10.4208/jcm.2101-m2020-0075
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