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Volume 38, Issue 1
An Error Analysis Method SPP-BEAM and a Construction Guideline of Nonconforming Finite Elements for Fourth Order Elliptic Problems

Jun Hu & Shangyou Zhang

DOI: 10.4208/jcm.1811-m2018-0162

J. Comp. Math., 38 (2020), pp. 195-222.

Published online: 2020-02

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  • Abstract

Under two hypotheses of nonconforming finite elements of fourth order elliptic problems, we present a side–patchwise projection based error analysis method (SPP–BEAM for short). Such a method is able to avoid both the regularity condition of exact solutions in the classical error analysis method and the complicated bubble function technique in the recent medius error analysis method. In addition, it is universal enough to admit generalizations. Then, we propose a sufficient condition for these hypotheses by imposing a set of in some sense necessary degrees of freedom of the shape function spaces. As an application, we use the theory to design a $P_3$ second order triangular $H^2$ non-conforming element by enriching two $P_4$ bubble functions and, another $P_4$ second order triangular $H^2$ nonconforming finite element, and a $P_3$ second order tetrahedral $H^2$ non-conforming element by enriching eight $P_4$ bubble functions, adding some more degrees of freedom.

  • Keywords

Nonconforming finite element, A priori error analysis, Biharmonic equation.

  • AMS Subject Headings

65N30, 73C02

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

hujun@math.pku.edu.cn (Jun Hu)

szhang@udel.edu (Shangyou Zhang)

  • BibTex
  • RIS
  • TXT
@Article{JCM-38-195, author = {Hu , Jun and Zhang , Shangyou}, title = {An Error Analysis Method SPP-BEAM and a Construction Guideline of Nonconforming Finite Elements for Fourth Order Elliptic Problems}, journal = {Journal of Computational Mathematics}, year = {2020}, volume = {38}, number = {1}, pages = {195--222}, abstract = {

Under two hypotheses of nonconforming finite elements of fourth order elliptic problems, we present a side–patchwise projection based error analysis method (SPP–BEAM for short). Such a method is able to avoid both the regularity condition of exact solutions in the classical error analysis method and the complicated bubble function technique in the recent medius error analysis method. In addition, it is universal enough to admit generalizations. Then, we propose a sufficient condition for these hypotheses by imposing a set of in some sense necessary degrees of freedom of the shape function spaces. As an application, we use the theory to design a $P_3$ second order triangular $H^2$ non-conforming element by enriching two $P_4$ bubble functions and, another $P_4$ second order triangular $H^2$ nonconforming finite element, and a $P_3$ second order tetrahedral $H^2$ non-conforming element by enriching eight $P_4$ bubble functions, adding some more degrees of freedom.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1811-m2018-0162}, url = {http://global-sci.org/intro/article_detail/jcm/13691.html} }
TY - JOUR T1 - An Error Analysis Method SPP-BEAM and a Construction Guideline of Nonconforming Finite Elements for Fourth Order Elliptic Problems AU - Hu , Jun AU - Zhang , Shangyou JO - Journal of Computational Mathematics VL - 1 SP - 195 EP - 222 PY - 2020 DA - 2020/02 SN - 38 DO - http://doi.org/10.4208/jcm.1811-m2018-0162 UR - https://global-sci.org/intro/article_detail/jcm/13691.html KW - Nonconforming finite element, A priori error analysis, Biharmonic equation. AB -

Under two hypotheses of nonconforming finite elements of fourth order elliptic problems, we present a side–patchwise projection based error analysis method (SPP–BEAM for short). Such a method is able to avoid both the regularity condition of exact solutions in the classical error analysis method and the complicated bubble function technique in the recent medius error analysis method. In addition, it is universal enough to admit generalizations. Then, we propose a sufficient condition for these hypotheses by imposing a set of in some sense necessary degrees of freedom of the shape function spaces. As an application, we use the theory to design a $P_3$ second order triangular $H^2$ non-conforming element by enriching two $P_4$ bubble functions and, another $P_4$ second order triangular $H^2$ nonconforming finite element, and a $P_3$ second order tetrahedral $H^2$ non-conforming element by enriching eight $P_4$ bubble functions, adding some more degrees of freedom.

Jun Hu & Shangyou Zhang. (2020). An Error Analysis Method SPP-BEAM and a Construction Guideline of Nonconforming Finite Elements for Fourth Order Elliptic Problems. Journal of Computational Mathematics. 38 (1). 195-222. doi:10.4208/jcm.1811-m2018-0162
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