Volume 13, Issue 1
A Two-Grid Finite-Volume Method for the Schrödinger Equation

Adv. Appl. Math. Mech., 13 (2021), pp. 176-190.

Published online: 2020-10

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• Abstract

In this paper, some two-grid finite-volume methods are constructed for solving the steady-state Schrödinger equation. The method projects the original coupled problem onto a coarser grid, on which it is less expensive to solve, and then prolongates the approximated coarse solution back to the fine grid, on which it is not much more difficult to solve the decoupled problem. We have shown, both theoretically and numerically,  that our schemes are more efficient and achieve asymptotically optimal accuracy as long as the mesh sizes satisfy $h=\mathcal{O}(H^2)$.

• Keywords

Schrödinger equation, coupled equation, finite volume, two-grid.

65N50, 65N30

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@Article{AAMM-13-176, author = {Hongmei Zhang , and Jianghua Yin , and Jicheng Jin , }, title = {A Two-Grid Finite-Volume Method for the Schrödinger Equation}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2020}, volume = {13}, number = {1}, pages = {176--190}, abstract = {

In this paper, some two-grid finite-volume methods are constructed for solving the steady-state Schrödinger equation. The method projects the original coupled problem onto a coarser grid, on which it is less expensive to solve, and then prolongates the approximated coarse solution back to the fine grid, on which it is not much more difficult to solve the decoupled problem. We have shown, both theoretically and numerically,  that our schemes are more efficient and achieve asymptotically optimal accuracy as long as the mesh sizes satisfy $h=\mathcal{O}(H^2)$.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2019-0212}, url = {http://global-sci.org/intro/article_detail/aamm/18346.html} }
TY - JOUR T1 - A Two-Grid Finite-Volume Method for the Schrödinger Equation AU - Hongmei Zhang , AU - Jianghua Yin , AU - Jicheng Jin , JO - Advances in Applied Mathematics and Mechanics VL - 1 SP - 176 EP - 190 PY - 2020 DA - 2020/10 SN - 13 DO - http://doi.org/10.4208/aamm.OA-2019-0212 UR - https://global-sci.org/intro/article_detail/aamm/18346.html KW - Schrödinger equation, coupled equation, finite volume, two-grid. AB -

In this paper, some two-grid finite-volume methods are constructed for solving the steady-state Schrödinger equation. The method projects the original coupled problem onto a coarser grid, on which it is less expensive to solve, and then prolongates the approximated coarse solution back to the fine grid, on which it is not much more difficult to solve the decoupled problem. We have shown, both theoretically and numerically,  that our schemes are more efficient and achieve asymptotically optimal accuracy as long as the mesh sizes satisfy $h=\mathcal{O}(H^2)$.

Hongmei Zhang, Jianghua Yin & Jicheng Jin. (2020). A Two-Grid Finite-Volume Method for the Schrödinger Equation. Advances in Applied Mathematics and Mechanics. 13 (1). 176-190. doi:10.4208/aamm.OA-2019-0212
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