Finite element method with discrete transparent boundary conditions for the time-dependent 1D Schrödinger equation

Alexander Zlotnik; Ilya Zlotnik

doi:10.3934/krm.2012.5.639

Article Contents

2012, Volume 5, Issue 3: 639-667. Doi: 10.3934/krm.2012.5.639

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Finite element method with discrete transparent boundary conditions for the time-dependent 1D Schrödinger equation

Alexander Zlotnik^1, and
Ilya Zlotnik^2,

1.
Department of Mathematics at Faculty of Economics Sciences, National Research University Higher School of Economics, Myasnitskaya 20, 101000 Moscow
2.
Department of Mathematical Modelling, Moscow Power Engineering Institute, Krasnokazarmennaya 14, 111250 Moscow

Received: February 29, 2012

Revised: April 30, 2012

Published: August 2012

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Abstract

We consider the time-dependent 1D Schrödinger equation on the half-axis with variable coefficients becoming constant for large $x$. We study a two-level symmetric in time (i.e. the Crank-Nicolson) and any order finite element in space numerical method to solve it. The method is coupled to an approximate transparent boundary condition (TBC). We prove uniform in time stability with respect to initial data and a free term in two norms, under suitable conditions on an operator in the approximate TBC. We also consider the corresponding method on an infinite mesh on the half-axis. We derive explicitly the discrete TBC allowing us to restrict the latter method to a finite mesh. The operator in the discrete TBC is a discrete convolution in time; in turn its kernel is a multiple discrete convolution. The stability conditions are justified for it. The accomplished computations confirm that high order finite elements coupled to the discrete TBC are effective even in the case of highly oscillating solutions and discontinuous potentials.

Keywords:

The time-dependent Schrödinger equation,
Galerkin method,
finite element method,
stability,
approximate and discrete transparent boundary conditions,
discrete convolution operator,
the tunnel effect.

Mathematics Subject Classification: 65M60, 65M12, 65M06, 35Q40.

Citation:

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References

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