TY - JOUR
T1 - Unconditional Error Analysis of VEMs for a Generalized Nonlinear Schrödinger Equation
AU - Li , Meng
AU - Zhao , Jikun
AU - Chen , Shaochun
JO - Journal of Computational Mathematics
VL - 2
SP - 500
EP - 543
PY - 2024
DA - 2024/01
SN - 42
DO - http://doi.org/10.4208/jcm.2207-m2022-0055
UR - https://global-sci.org/intro/article_detail/jcm/22890.html
KW - Conforming and nonconforming, Virtual element methods, Leap-frog scheme, Generalized nonlinear Schrödinger system, Unconditionally optimal error estimates.
AB - <p style="text-align: justify;">In this work, we focus on the conforming and nonconforming leap-frog virtual element
methods for the generalized nonlinear Schrödinger equation, and establish their unconditional stability and optimal error estimates. By constructing a time-discrete system, the
error between the solutions of the continuous model and the numerical scheme is separated
into the temporal error and the spatial error, which makes the spatial error $\tau$-independent.
The inverse inequalities in the existing conforming and new constructed nonconforming
virtual element spaces are utilized to derive the $L^∞$-norm uniform boundedness of numerical solutions without any restrictions on time-space step ratio, and then unconditionally
optimal error estimates of the numerical schemes are obtained naturally. What needs to
be emphasized is that if we use the pre-existing nonconforming virtual elements, there is
no way to derive the $L^∞$-norm uniform boundedness of the functions in the nonconforming virtual element spaces so as to be hard to get the corresponding inverse inequalities.
Finally, several numerical examples are reported to confirm our theoretical results.</p>