TY - JOUR
T1 - An Efficient and Unconditionally Convergent Galerkin Finite Element Method for the Nonlinear Schrödinger Equation in High Dimensions
AU - Cheng , Yue
AU - Wang , Tingchun
AU - Guo , Boling
JO - Advances in Applied Mathematics and Mechanics
VL - 4
SP - 735
EP - 760
PY - 2021
DA - 2021/04
SN - 13
DO - http://doi.org/10.4208/aamm.OA-2020-0033
UR - https://global-sci.org/intro/article_detail/aamm/18749.html
KW - Nonlinear Schrödinger  equation, linearized Galerkin FEM, unconditional convergence, optimal error estimate.
AB - <p style="text-align: justify;">In this paper, we aim to propose and analyze a linearized three-level Galerkin finite element method (FEM) for the nonlinear Schrödinger equation with a general nonlinearity and an external potential. Compared with the existing results in literature, under a weaker assumption on both the exact solution and the nonlinear term,&nbsp; we give a concise proof to establish the optimal $L^{2}$ error estimate without any grid-ratio restriction.&nbsp; Besides the standard energy method, the key tools used in our analysis are an induction argument and several Sobolev inequalities. Numerical results are reported to verify our theoretical analysis.</p>