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 - <p style="text-align: justify;">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.</p>