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Volume 17, Issue 3
An Extended Courant Element on a Polytope with Application in Approximating an Obstacle Problem

Mingqing Chen & Jianguo Huang

DOI: 10.4208/nmtma.OA-2024-0001

Numer. Math. Theor. Meth. Appl., 17 (2024), pp. 607-629.

Published online: 2024-08

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  • Abstract

An extended Courant element is constructed on an $n$ dimensional polytope $K,$ which reduces to the usual Courant element when $K$ is a simplex. The set of the degrees of freedom consists of function values at all vertices of $K,$ while the shape function space $P_K$ is formed by repeatedly using the harmonic extension from lower dimensional face to higher dimensional face. Several fundamental estimates are derived on this element under reasonable geometric assumptions. Moreover, the weak maximum principle holds for any function in $P_K,$ which enables us to use the element for approximating an obstacle problem in three dimensions. The corresponding optimal error estimate in $H^1$-norm is also established. Numerical results are reported to verify theoretical findings.

  • Keywords

Extended Courant element, virtual element, quasi-elliptic projection operators, obstacle problem.

  • AMS Subject Headings

65N30, 65N12, 65K15

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-17-607, author = {Chen , Mingqing and Huang , Jianguo}, title = {An Extended Courant Element on a Polytope with Application in Approximating an Obstacle Problem}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2024}, volume = {17}, number = {3}, pages = {607--629}, abstract = {

An extended Courant element is constructed on an $n$ dimensional polytope $K,$ which reduces to the usual Courant element when $K$ is a simplex. The set of the degrees of freedom consists of function values at all vertices of $K,$ while the shape function space $P_K$ is formed by repeatedly using the harmonic extension from lower dimensional face to higher dimensional face. Several fundamental estimates are derived on this element under reasonable geometric assumptions. Moreover, the weak maximum principle holds for any function in $P_K,$ which enables us to use the element for approximating an obstacle problem in three dimensions. The corresponding optimal error estimate in $H^1$-norm is also established. Numerical results are reported to verify theoretical findings.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2024-0001}, url = {http://global-sci.org/intro/article_detail/nmtma/23368.html} }
TY - JOUR T1 - An Extended Courant Element on a Polytope with Application in Approximating an Obstacle Problem AU - Chen , Mingqing AU - Huang , Jianguo JO - Numerical Mathematics: Theory, Methods and Applications VL - 3 SP - 607 EP - 629 PY - 2024 DA - 2024/08 SN - 17 DO - http://doi.org/10.4208/nmtma.OA-2024-0001 UR - https://global-sci.org/intro/article_detail/nmtma/23368.html KW - Extended Courant element, virtual element, quasi-elliptic projection operators, obstacle problem. AB -

An extended Courant element is constructed on an $n$ dimensional polytope $K,$ which reduces to the usual Courant element when $K$ is a simplex. The set of the degrees of freedom consists of function values at all vertices of $K,$ while the shape function space $P_K$ is formed by repeatedly using the harmonic extension from lower dimensional face to higher dimensional face. Several fundamental estimates are derived on this element under reasonable geometric assumptions. Moreover, the weak maximum principle holds for any function in $P_K,$ which enables us to use the element for approximating an obstacle problem in three dimensions. The corresponding optimal error estimate in $H^1$-norm is also established. Numerical results are reported to verify theoretical findings.

Chen , Mingqing and Huang , Jianguo. (2024). An Extended Courant Element on a Polytope with Application in Approximating an Obstacle Problem. Numerical Mathematics: Theory, Methods and Applications. 17 (3). 607-629. doi:10.4208/nmtma.OA-2024-0001
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