A Preconditioned Iterative Solver for the Scattering Solutions of the Schrodinger Equation
Hisham bin Zubair 1*, Bram Reps 2, Wim Vanroose 21 Department of Mathematical Sciences, Faculty of Computer Science, Institute of Business Administration, University Rd., 75270 Karachi, Pakistan.
2 Department of Mathematics and Computer Science, University of Antwerp, Middelheimlaan 1, B-2020 Antwerpen, Belgium.
Received 12 December 2009; Accepted (in revised version) 18 September 2010
Available online 24 October 2011
The Schrodinger equation defines the dynamics of quantum particles which has been an area of unabated interest in physics. We demonstrate how simple transformations of the Schrodinger equation leads to a coupled linear system, whereby each diagonal block is a high frequency Helmholtz problem. Based on this model, we derive indefinite Helmholtz model problems with strongly varying wavenumbers. We employ the iterative approach for their solution. In particular, we develop a preconditioner that has its spectrum restricted to a quadrant (of the complex plane) thereby making it easily invertible by multigrid methods with standard components. This multigrid preconditioner is used in conjunction with suitable Krylov-subspace methods for solving the indefinite Helmholtz model problems. The aim of this study is to report the feasibility of this preconditioner for the model problems. We compare this idea with the other prevalent preconditioning ideas, and discuss its merits. Results of numerical experiments are presented, which complement the proposed ideas, and show that this preconditioner may be used in an automatic setting.AMS subject classifications: 65F08, 65F10, 65F15, 65M55
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Key words: Scattering, Schrodinger equation, exterior complex scaling, preconditioning, multigrid, complex-shifted Laplacian (CSL), complex-scaled grid (CSG), quadrant-definite (QD).
Email: firstname.lastname@example.org (H. bin Zubair), email@example.com (B. Reps), firstname.lastname@example.org (W. Vanroose)