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Volume 8, Issue 5
Coefficient Jump-Independent Approximation of the Conforming and Nonconforming Finite Element Solutions

Shangyou Zhang

DOI: 10.4208/aamm.2015.m931

Adv. Appl. Math. Mech., 8 (2016), pp. 722-736.

Published online: 2018-05

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  • Abstract

A counterexample is constructed. It confirms that the error of conforming finite element solution is proportional to the coefficient jump, when solving interface elliptic equations. The Scott-Zhang operator is applied to a nonconforming finite element. It is shown that the nonconforming finite element provides the optimal order approximation in interpolation, in $L^2$-projection, and in solving elliptic differential equation, independent of the coefficient jump in the elliptic differential equation. Numerical tests confirm the theoretical finding.

  • Keywords

Jump coefficient, finite element, $L^2$ projection, weighted projection, Scott-Zhang operator.

  • AMS Subject Headings

65N30, 65D05

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{AAMM-8-722, author = {Zhang , Shangyou}, title = {Coefficient Jump-Independent Approximation of the Conforming and Nonconforming Finite Element Solutions}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2018}, volume = {8}, number = {5}, pages = {722--736}, abstract = {

A counterexample is constructed. It confirms that the error of conforming finite element solution is proportional to the coefficient jump, when solving interface elliptic equations. The Scott-Zhang operator is applied to a nonconforming finite element. It is shown that the nonconforming finite element provides the optimal order approximation in interpolation, in $L^2$-projection, and in solving elliptic differential equation, independent of the coefficient jump in the elliptic differential equation. Numerical tests confirm the theoretical finding.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.2015.m931}, url = {http://global-sci.org/intro/article_detail/aamm/12112.html} }
TY - JOUR T1 - Coefficient Jump-Independent Approximation of the Conforming and Nonconforming Finite Element Solutions AU - Zhang , Shangyou JO - Advances in Applied Mathematics and Mechanics VL - 5 SP - 722 EP - 736 PY - 2018 DA - 2018/05 SN - 8 DO - http://doi.org/10.4208/aamm.2015.m931 UR - https://global-sci.org/intro/article_detail/aamm/12112.html KW - Jump coefficient, finite element, $L^2$ projection, weighted projection, Scott-Zhang operator. AB -

A counterexample is constructed. It confirms that the error of conforming finite element solution is proportional to the coefficient jump, when solving interface elliptic equations. The Scott-Zhang operator is applied to a nonconforming finite element. It is shown that the nonconforming finite element provides the optimal order approximation in interpolation, in $L^2$-projection, and in solving elliptic differential equation, independent of the coefficient jump in the elliptic differential equation. Numerical tests confirm the theoretical finding.

Shangyou Zhang. (2020). Coefficient Jump-Independent Approximation of the Conforming and Nonconforming Finite Element Solutions. Advances in Applied Mathematics and Mechanics. 8 (5). 722-736. doi:10.4208/aamm.2015.m931
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