Kinetic and Related Models (KRM)

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

Pages: 639 - 667, Volume 5, Issue 3, September 2012      doi:10.3934/krm.2012.5.639

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Alexander Zlotnik - Department of Mathematics at Faculty of Economics Sciences, National Research University Higher School of Economics, Myasnitskaya 20, 101000 Moscow, Russian Federation (email)
Ilya Zlotnik - Department of Mathematical Modelling, Moscow Power Engineering Institute, Krasnokazarmennaya 14, 111250 Moscow, Russian Federation (email)

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.

Received: March 2012;      Revised: May 2012;      Available Online: August 2012.