Finite element method for an eigenvalue optimization problem of the Schrödinger operator

Shuangbing Guo; Xiliang Lu; Zhiyue Zhang; Shuangbing Guo; Xiliang Lu; Zhiyue Zhang

doi:10.3934/math.2022281

AIMS Mathematics

2022, Volume 7, Issue 4: 5049-5071. doi: 10.3934/math.2022281

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Research article

Finite element method for an eigenvalue optimization problem of the Schrödinger operator

1.
School of Mathematical Science, Henan Institute of Science and Technology, Xinxiang, 453003, China
2.
School of Mathematical Science, Nanjing Normal University, Nanjing, 210023, China
3.
School of Mathematics and Statistics, and Hubei Key Laboratory of Computational Science, Wuhan University, Wuhan, 430072, China

Received: 26 August 2021 Revised: 11 December 2021 Accepted: 19 December 2021 Published: 29 December 2021
MSC : 35J10, 35P15, 49Q10, 65N25, 65N30

In this paper, we study the optimization algorithm to compute the smallest eigenvalue of the Schrödinger operator with volume constraint. A finite element discretization of this problem is established. We provide the error estimate for the numerical solution. The optimal solution can be approximated by a fixed point iteration scheme. Then a monotonic decreasing algorithm is presented to solve the eigenvalue optimization problem. Numerical simulations demonstrate the efficiency of the method.
- eigenvalue optimization,
- Schrödinger operator,
- shape optimization,
- finite element method,
- error estimate
Citation: Shuangbing Guo, Xiliang Lu, Zhiyue Zhang. Finite element method for an eigenvalue optimization problem of the Schrödinger operator[J]. AIMS Mathematics, 2022, 7(4): 5049-5071. doi: 10.3934/math.2022281

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Abstract

In this paper, we study the optimization algorithm to compute the smallest eigenvalue of the Schrödinger operator with volume constraint. A finite element discretization of this problem is established. We provide the error estimate for the numerical solution. The optimal solution can be approximated by a fixed point iteration scheme. Then a monotonic decreasing algorithm is presented to solve the eigenvalue optimization problem. Numerical simulations demonstrate the efficiency of the method.

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