TY - JOUR
T1 - Nodal $\mathcal{O}(h^4)$-Superconvergence in 3D by Averaging Piecewise Linear, Bilinear, and Trilinear FE Approximations
AU - Antti Hannukainen, Sergey Korotov & Michal Křížek
JO - Journal of Computational Mathematics
VL - 1
SP - 1
EP - 10
PY - 2010
DA - 2010/02
SN - 28
DO - http://doi.org/10.4208/jcm.2009.09-m1004
UR - https://global-sci.org/intro/article_detail/jcm/8503.html
KW - Higher order error estimates, Tetrahedral and prismatic elements, Superconvergence, Averaging operators.
AB - <p style="text-align: justify;">We construct and analyse a nodal $\mathcal{O}(h^4)$-superconvergent FE scheme for approximating the Poisson equation with homogeneous boundary conditions in three-dimensional domains by means of piecewise trilinear functions. The scheme is based on averaging the equations that arise from FE approximations on uniform cubic, tetrahedral, and prismatic partitions. This approach presents a three-dimensional generalization of a two-dimensional averaging of linear and bilinear elements which also exhibits nodal $\mathcal{O}(h^4)$-superconvergence (ultraconvergence). The obtained superconvergence result is illustrated by two numerical examples.</p>