TY - JOUR
T1 - Superconvergence of Finite Element Approximations of the Two-Dimensional Cubic Nonlinear Schrödinger Equation
AU - Wang , Jianyun
AU - Tian , Zhikun
JO - Advances in Applied Mathematics and Mechanics
VL - 3
SP - 652
EP - 665
PY - 2022
DA - 2022/02
SN - 14
DO - http://doi.org/10.4208/aamm.OA-2020-0268
UR - https://global-sci.org/intro/article_detail/aamm/20279.html
KW - Superconvergence, nonlinear Schrödinger equation, finite element method, elliptic projection.
AB - <p style="text-align: justify;">The superconvergence of a two-dimensional time-independent nonlinear
Schrödinger equation are analyzed with the rectangular Lagrange type finite element of order $k$. Firstly, the error estimate and superclose property are given in $H^1$-norm
with order $\mathcal{O}(h^{k+1})$ between the finite element solution $u_h$ and the interpolation function $u_I$ by use of the elliptic projection operator. Then, the global superconvergence is
obtained by the interpolation post-processing technique. In addition, some numerical
examples with the order $k = 1$ and $k = 2$ are provided to demonstrate the theoretical
analysis.</p>