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Volume 37, Issue 4
Interior Estimates of Semidiscrete Finite Element Methods for Parabolic Problems with Distributional Data

Li Guo, Hengguang Li & Yang Yang

DOI: 10.4208/jcm.1804-m2017-0240

J. Comp. Math., 37 (2019), pp. 458-474.

Published online: 2019-02

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  • Abstract

Let $Ω ⊂ \mathbb{R}^d$, $1 ≤ d ≤ 3$, be a bounded $d$-polytope. Consider the parabolic equation on $Ω$ with the Dirac delta function on the right hand side. We study the well-posedness, regularity, and the interior error estimate of semidiscrete finite element approximations of the equation. In particular, we derive that the interior error is bounded by the best local approximation error, the negative norms of the error, and the negative norms of the time derivative of the error. This result implies different convergence rates for the numerical solution in different interior regions, especially when the region is close to the singular point. Numerical test results are reported to support the theoretical prediction.

  • Keywords

Parabolic problems, Distributional data, Finite element methods, Interior estimates, Well-posedness, Singularity.

  • AMS Subject Headings

65M15, 65M60

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

guoli7@mail.sysu.edu.cn (Li Guo)

li@wayne.edu (Hengguang Li)

yyang7@mtu.edu (Yang Yang)

  • BibTex
  • RIS
  • TXT
@Article{JCM-37-458, author = {Guo , LiLi , Hengguang and Yang , Yang}, title = {Interior Estimates of Semidiscrete Finite Element Methods for Parabolic Problems with Distributional Data}, journal = {Journal of Computational Mathematics}, year = {2019}, volume = {37}, number = {4}, pages = {458--474}, abstract = {

Let $Ω ⊂ \mathbb{R}^d$, $1 ≤ d ≤ 3$, be a bounded $d$-polytope. Consider the parabolic equation on $Ω$ with the Dirac delta function on the right hand side. We study the well-posedness, regularity, and the interior error estimate of semidiscrete finite element approximations of the equation. In particular, we derive that the interior error is bounded by the best local approximation error, the negative norms of the error, and the negative norms of the time derivative of the error. This result implies different convergence rates for the numerical solution in different interior regions, especially when the region is close to the singular point. Numerical test results are reported to support the theoretical prediction.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1804-m2017-0240}, url = {http://global-sci.org/intro/article_detail/jcm/13002.html} }
TY - JOUR T1 - Interior Estimates of Semidiscrete Finite Element Methods for Parabolic Problems with Distributional Data AU - Guo , Li AU - Li , Hengguang AU - Yang , Yang JO - Journal of Computational Mathematics VL - 4 SP - 458 EP - 474 PY - 2019 DA - 2019/02 SN - 37 DO - http://doi.org/10.4208/jcm.1804-m2017-0240 UR - https://global-sci.org/intro/article_detail/jcm/13002.html KW - Parabolic problems, Distributional data, Finite element methods, Interior estimates, Well-posedness, Singularity. AB -

Let $Ω ⊂ \mathbb{R}^d$, $1 ≤ d ≤ 3$, be a bounded $d$-polytope. Consider the parabolic equation on $Ω$ with the Dirac delta function on the right hand side. We study the well-posedness, regularity, and the interior error estimate of semidiscrete finite element approximations of the equation. In particular, we derive that the interior error is bounded by the best local approximation error, the negative norms of the error, and the negative norms of the time derivative of the error. This result implies different convergence rates for the numerical solution in different interior regions, especially when the region is close to the singular point. Numerical test results are reported to support the theoretical prediction.

Guo , LiLi , Hengguang and Yang , Yang. (2019). Interior Estimates of Semidiscrete Finite Element Methods for Parabolic Problems with Distributional Data. Journal of Computational Mathematics. 37 (4). 458-474. doi:10.4208/jcm.1804-m2017-0240
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