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Research Article | Open Access

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

Shuangbing Guo1,2( ), Xiliang Lu3, Zhiyue Zhang2
School of Mathematical Science, Henan Institute of Science and Technology, Xinxiang, 453003, China
School of Mathematical Science, Nanjing Normal University, Nanjing, 210023, China
School of Mathematics and Statistics, and Hubei Key Laboratory of Computational Science, Wuhan University, Wuhan, 430072, China
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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.

Keywords

eigenvalue optimizationSchrödinger operatorshape optimizationfinite element methoderror estimate
CLC number: 35J10, 35P15, 49Q10, 65N25, 65N30

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AIMS Mathematics
Volume 7 Issue 4,
April 2022
Pages 5049-5071
DOI: 10.3934/math.2022281

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Cite this article:
Guo S, Lu X, Zhang Z. Finite element method for an eigenvalue optimization problem of the Schrödinger operator. AIMS Mathematics, 2022, 7(4): 5049-5071. https://doi.org/10.3934/math.2022281

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Received: 26 August 2021
Revised: 11 December 2021
Accepted: 19 December 2021
Published: 15 April 2022
©2022 the Author(s), licensee AIMS Press.

This is an open access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0)