TY - JOUR
T1 - Unconditionally Superclose Analysis of a New Mixed Finite Element Method for Nonlinear Parabolic Equation
AU - Shi , Dongyang
AU - Yan , Fengna
AU - Wang , Junjun
JO - Journal of Computational Mathematics
VL - 1
SP - 1
EP - 17
PY - 2018
DA - 2018/08
SN - 37
DO - http://doi.org/10.4208/jcm.1707-m2016-0718
UR - https://global-sci.org/intro/article_detail/jcm/12645.html
KW - Nonlinear parabolic equation, Mixed FEM, Time-discrete and spatial-discrete
systems, $τ$-independent superclose results.
AB - <p style="text-align: justify;">This paper develops a framework to deal with the unconditional superclose analysis of
nonlinear parabolic equation. Taking the finite element pair $Q_{11}/Q_{01} × Q_{10}$ as an example,
a new mixed finite element method (FEM) is established and the $τ$ -independent superclose
results of the original variable $u$ in $H^1$-norm and the flux variable $\mathop{q} \limits ^{\rightarrow}= −a(u)∇u$ in $L^2$-norm are deduced ($τ$ is the temporal partition parameter). A key to our analysis is an
error splitting technique, with which the time-discrete and the spatial-discrete systems are
constructed, respectively. For the first system, the boundedness of the temporal errors is obtained. For the second system, the spatial superclose results are presented unconditionally, while the previous literature always only obtain the convergent estimates or require
certain time step conditions. Finally, some numerical results are provided to confirm the
theoretical analysis, and show the efficiency of the proposed method.</p>