An adaptive FEM with ITP approach for steady Schrödinger equation

ABSTRACT

In this paper, an adaptive numerical method is proposed for solving a 2D Schrödinger equation with an imaginary time propagation approach. The differential equation is first transferred via a Wick rotation to a real time-dependent equation, whose solution corresponds to the ground state of a given system when time approaches infinity. The temporal equation is then discretized spatially via a finite element method, and temporally utilizing a Crank–Nicolson scheme. A moving mesh strategy based on harmonic maps is considered to eliminate possible singular behaviour of the solution. Several linear and nonlinear examples are tested by using our method. The experiments demonstrate clearly that our method provides an effective way to locate the ground state of the equations through underlying eigenvalue problems.

KEYWORDS:

• Schrödinger equation
• imaginary time propagation
• moving mesh method
• finite element method
• ground state

2010 AMS SUBJECT CLASSIFICATIONS:

• 65N30
• 35Q40

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The first author would like to thank the support from Postgraduate Studentship from University of Macau. The work of the second author is partially supported by Fundo para o Desenvolvimento das Ciências e da Tecnologia (FDCT) 029/2016/A1, 050/2014/A1 from Macao SAR, MYRG2017-00189-FST, MYRG2014-00109-FST from University of Macau, and National Natural Science Foundation of China [Grant No. 11401608].