Volume 13, Issue 4
An Efficient and Unconditionally Convergent Galerkin Finite Element Method for the Nonlinear Schrödinger Equation in High Dimensions

Adv. Appl. Math. Mech., 13 (2021), pp. 735-760.

Published online: 2021-04

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• Abstract

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,  we give a concise proof to establish the optimal $L^{2}$ error estimate without any grid-ratio restriction.  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.

• Keywords

Nonlinear Schrödinger equation, linearized Galerkin FEM, unconditional convergence, optimal error estimate.

65N12, 65N15

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@Article{AAMM-13-735, author = {Yue and Cheng and and 14804 and and Yue Cheng and Tingchun and Wang and and 14805 and and Tingchun Wang and Boling and Guo and and 14806 and and Boling Guo}, title = {An Efficient and Unconditionally Convergent Galerkin Finite Element Method for the Nonlinear Schrödinger Equation in High Dimensions}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2021}, volume = {13}, number = {4}, pages = {735--760}, abstract = {

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,  we give a concise proof to establish the optimal $L^{2}$ error estimate without any grid-ratio restriction.  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.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2020-0033}, url = {http://global-sci.org/intro/article_detail/aamm/18749.html} }
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 -

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,  we give a concise proof to establish the optimal $L^{2}$ error estimate without any grid-ratio restriction.  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.

Yue Cheng, Tingchun Wang & Boling Guo. (1970). An Efficient and Unconditionally Convergent Galerkin Finite Element Method for the Nonlinear Schrödinger Equation in High Dimensions. Advances in Applied Mathematics and Mechanics. 13 (4). 735-760. doi:10.4208/aamm.OA-2020-0033
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