TY - JOUR
T1 - Conservative Conforming and Nonconforming VEMs for Fourth Order Nonlinear Schrödinger Equations with Trapped Term
AU - Li , Meng
AU - Zhao , Jikun
AU - Wang , Zhongchi
AU - Chen , Shaochun
JO - Journal of Computational Mathematics
VL - 2
SP - 454
EP - 499
PY - 2024
DA - 2024/01
SN - 42
DO - http://doi.org/10.4208/jcm.2209-m2021-0038
UR - https://global-sci.org/intro/article_detail/jcm/22889.html
KW - $H^2$ conforming virtual element, $C^0$ nonconforming virtual element, Morley-type nonconforming virtual element, Nonlinear Schrödinger equation, Conservation, Convergence.
AB - <p style="text-align: justify;">This paper aims to construct and analyze the conforming and nonconforming virtual
element methods for a class of fourth order nonlinear Schrödinger equations with trapped
term. We mainly consider three types of virtual elements, including $H^2$ conforming virtual element, $C^0$ nonconforming virtual element and Morley-type nonconforming virtual
element. The fully discrete schemes are constructed by virtue of virtual element methods
in space and modified Crank-Nicolson method in time. We prove the mass and energy
conservation, the boundedness and the unique solvability of the fully discrete schemes.
After introducing a new type of the Ritz projection, the optimal and unconditional error
estimates for the fully discrete schemes are presented and proved. Finally, two numerical
examples are investigated to confirm our theoretical analysis.</p>