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Volume 18, Issue 1
Pointwise Goal-Oriented a Posteriori Error Estimates Using Dual Problems with Dirac Delta Source Terms for Linear Elliptic Problems

Fei Li, Jingang Liu, Nianyu Yi & Liuqiang Zhong

DOI: 10.4208/aamm.OA-2023-0186

Adv. Appl. Math. Mech., 18 (2026), pp. 222-241.

Published online: 2025-10

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  • Abstract

In this paper, a pointwise goal-oriented residual-based a posteriori error estimator is proposed for linear elliptic equations with restricted source terms. The pointwise error is directly estimated by introducing the dual problem with a Dirac delta source term instead of using classical mollification technique. The goal-oriented error estimator is proved to be the upper bound of the pointwise error. Numerical experiments show the advantage of the adaptive finite element method (AFEM) based on this error estimator, which can preserve the monotonicity of the pointwise error, compared with the goal-oriented AFEM using the mollification technique.

  • Keywords

Pointwise quantity of interest, a posteriori error estimate, adaptive finite element method, Dirac delta source term.

  • AMS Subject Headings

65M10, 78A48

  • Copyright

COPYRIGHT: © Global Science Press

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  • TXT
@Article{AAMM-18-222, author = {Li , FeiLiu , JingangYi , Nianyu and Zhong , Liuqiang}, title = {Pointwise Goal-Oriented a Posteriori Error Estimates Using Dual Problems with Dirac Delta Source Terms for Linear Elliptic Problems}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2025}, volume = {18}, number = {1}, pages = {222--241}, abstract = {

In this paper, a pointwise goal-oriented residual-based a posteriori error estimator is proposed for linear elliptic equations with restricted source terms. The pointwise error is directly estimated by introducing the dual problem with a Dirac delta source term instead of using classical mollification technique. The goal-oriented error estimator is proved to be the upper bound of the pointwise error. Numerical experiments show the advantage of the adaptive finite element method (AFEM) based on this error estimator, which can preserve the monotonicity of the pointwise error, compared with the goal-oriented AFEM using the mollification technique.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2023-0186}, url = {http://global-sci.org/intro/article_detail/aamm/24525.html} }
TY - JOUR T1 - Pointwise Goal-Oriented a Posteriori Error Estimates Using Dual Problems with Dirac Delta Source Terms for Linear Elliptic Problems AU - Li , Fei AU - Liu , Jingang AU - Yi , Nianyu AU - Zhong , Liuqiang JO - Advances in Applied Mathematics and Mechanics VL - 1 SP - 222 EP - 241 PY - 2025 DA - 2025/10 SN - 18 DO - http://doi.org/10.4208/aamm.OA-2023-0186 UR - https://global-sci.org/intro/article_detail/aamm/24525.html KW - Pointwise quantity of interest, a posteriori error estimate, adaptive finite element method, Dirac delta source term. AB -

In this paper, a pointwise goal-oriented residual-based a posteriori error estimator is proposed for linear elliptic equations with restricted source terms. The pointwise error is directly estimated by introducing the dual problem with a Dirac delta source term instead of using classical mollification technique. The goal-oriented error estimator is proved to be the upper bound of the pointwise error. Numerical experiments show the advantage of the adaptive finite element method (AFEM) based on this error estimator, which can preserve the monotonicity of the pointwise error, compared with the goal-oriented AFEM using the mollification technique.

Li , FeiLiu , JingangYi , Nianyu and Zhong , Liuqiang. (2025). Pointwise Goal-Oriented a Posteriori Error Estimates Using Dual Problems with Dirac Delta Source Terms for Linear Elliptic Problems. Advances in Applied Mathematics and Mechanics. 18 (1). 222-241. doi:10.4208/aamm.OA-2023-0186
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