TY - JOUR
T1 - Locally Conservative Serendipity Finite Element Solutions for Elliptic Equations
AU - Zhou , Yanhui
AU - Zou , Qingsong
JO - International Journal of Numerical Analysis and Modeling
VL - 1
SP - 19
EP - 37
PY - 2021
DA - 2021/02
SN - 18
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/ijnam/18619.html
KW - Postprocessing, serendipity finite elements, local conservation laws, error estimates.
AB - <p style="text-align: justify;">In this paper, we post-process an eight-node-serendipity finite element solution for
elliptic equations. In the post-processing procedure, we first construct a $control$ $volume$ for each
node in the serendipity finite element mesh, then we enlarge the serendipity finite element space by
adding some appropriate element-wise bubbles and require the novel solution to satisfy the local
conservation law on each control volume. Our post-processing procedure can be implemented in
a parallel computing environment and its computational cost is proportional to the cardinality of
the serendipity elements. Moreover, both our theoretical analysis and numerical examples show
that the postprocessed solution converges to the exact solution with optimal convergence rates
both under $H^1$ and $L^2$ norms. A numerical experiment for a single-phase porous media problem
validates the necessity of the post-processing procedure.</p>