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Volume 28, Issue 2
Super-Closeness Between the Ritz Projection and the Finite Element Solution for Some Elliptic Problems

Ruosi Liang, Yue Yan, Wenbin Chen & Yanqiu Wang

DOI: 10.4208/cicp.OA-2019-0154

Commun. Comput. Phys., 28 (2020), pp. 803-826.

Published online: 2020-06

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  • Abstract

We prove the super-closeness between the finite element solution and the Ritz projection for some second order and fourth order elliptic equations in both the $H^1$ and the $L^2$ norms. For the fourth order problem, a Ciarlet-Raviart type mixed formulation is used in the analysis. The main tool in the proof is a negative norm estimate of the Ritz projection, which requires $H^{q+1}$ regularity for second order elliptic equations. Therefore, the analysis is done on a domain Ω with smooth boundary, and hence we only consider the pure Neumann boundary problems which can be discretized naturally on such domains, if ignoring the effect of numerical integrals. For the fourth order problem, our results amend the gap between the theoretical estimates and the numerical examples in a previous work [22].

  • Keywords

Finite element, Ritz projection, negative norm estimate, super-closeness estimate.

  • AMS Subject Headings

65N12, 65N30, 35J30

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

wbchen@fudan.edu.cn (Wenbin Chen)

  • BibTex
  • RIS
  • TXT
@Article{CiCP-28-803, author = {Liang , RuosiYan , YueChen , Wenbin and Wang , Yanqiu}, title = {Super-Closeness Between the Ritz Projection and the Finite Element Solution for Some Elliptic Problems}, journal = {Communications in Computational Physics}, year = {2020}, volume = {28}, number = {2}, pages = {803--826}, abstract = {

We prove the super-closeness between the finite element solution and the Ritz projection for some second order and fourth order elliptic equations in both the $H^1$ and the $L^2$ norms. For the fourth order problem, a Ciarlet-Raviart type mixed formulation is used in the analysis. The main tool in the proof is a negative norm estimate of the Ritz projection, which requires $H^{q+1}$ regularity for second order elliptic equations. Therefore, the analysis is done on a domain Ω with smooth boundary, and hence we only consider the pure Neumann boundary problems which can be discretized naturally on such domains, if ignoring the effect of numerical integrals. For the fourth order problem, our results amend the gap between the theoretical estimates and the numerical examples in a previous work [22].

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2019-0154}, url = {http://global-sci.org/intro/article_detail/cicp/16954.html} }
TY - JOUR T1 - Super-Closeness Between the Ritz Projection and the Finite Element Solution for Some Elliptic Problems AU - Liang , Ruosi AU - Yan , Yue AU - Chen , Wenbin AU - Wang , Yanqiu JO - Communications in Computational Physics VL - 2 SP - 803 EP - 826 PY - 2020 DA - 2020/06 SN - 28 DO - http://doi.org/10.4208/cicp.OA-2019-0154 UR - https://global-sci.org/intro/article_detail/cicp/16954.html KW - Finite element, Ritz projection, negative norm estimate, super-closeness estimate. AB -

We prove the super-closeness between the finite element solution and the Ritz projection for some second order and fourth order elliptic equations in both the $H^1$ and the $L^2$ norms. For the fourth order problem, a Ciarlet-Raviart type mixed formulation is used in the analysis. The main tool in the proof is a negative norm estimate of the Ritz projection, which requires $H^{q+1}$ regularity for second order elliptic equations. Therefore, the analysis is done on a domain Ω with smooth boundary, and hence we only consider the pure Neumann boundary problems which can be discretized naturally on such domains, if ignoring the effect of numerical integrals. For the fourth order problem, our results amend the gap between the theoretical estimates and the numerical examples in a previous work [22].

Ruosi Liang, Yue Yan, Wenbin Chen & Yanqiu Wang. (2020). Super-Closeness Between the Ritz Projection and the Finite Element Solution for Some Elliptic Problems. Communications in Computational Physics. 28 (2). 803-826. doi:10.4208/cicp.OA-2019-0154
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