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Two‐grid method for two‐dimensional nonlinear Schrödinger equation by finite element method
Author(s) -
Hu Hanzhang
Publication year - 2018
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.22193
Subject(s) - grid , mathematics , finite element method , nonlinear system , scheme (mathematics) , space (punctuation) , mathematical analysis , geometry , computer science , physics , quantum mechanics , thermodynamics , operating system
A conservative two‐grid finite element scheme is presented for the two‐dimensional nonlinear Schrödinger equation. One Newton iteration is applied on the fine grid to linearize the fully discrete problem using the coarse‐grid solution as the initial guess. Moreover, error estimates are conducted for the two‐grid method. It is shown that the coarse space can be extremely coarse, with no loss in the order of accuracy, and still achieve the asymptotically optimal approximation as long as the mesh sizes satisfy H = O ( h1 2) in the two‐grid method. The numerical results show that this method is very effective.