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Crank-Nicolson finite element discretizations for a two-dimensional linear Schrödinger-type equation posed in a noncylindrical domain


Authors: D. C. Antonopoulou, G. D. Karali, M. Plexousakis and G. E. Zouraris
Journal: Math. Comp. 84 (2015), 1571-1598
MSC (2000): Primary 65M12, 65M15, 65M60
DOI: https://doi.org/10.1090/S0025-5718-2014-02900-1
Published electronically: November 5, 2014
MathSciNet review: 3335884
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Abstract: Motivated by the paraxial narrow-angle approximation of the Helmholtz equation in domains of variable topography, we consider an initial- and boundary-value problem for a general Schrödinger-type equation posed on a two space-dimensional noncylindrical domain with mixed boundary conditions. The problem is transformed into an equivalent one posed on a rectangular domain, and we approximate its solution by a Crank-Nicolson finite element method. For the proposed numerical method, we derive an optimal order error estimate in the $ L^2$ norm, and to support the error analysis we prove a global elliptic regularity theorem for complex elliptic boundary value problems with mixed boundary conditions. Results from numerical experiments are presented which verify the optimal order of convergence of the method.


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Additional Information

D. C. Antonopoulou
Affiliation: Department of Mathematics and Applied Mathematics, University of Crete, P.O. Box 2208, GR-710 03 Heraklion, Crete, Greece – and – Institute of Applied and Computational Mathematics, FORTH, P.O. Box 1527, GR-711 10 Heraklion, Crete, Greece
Email: danton@tem.uoc.gr

G. D. Karali
Affiliation: Department of Mathematics and Applied Mathematics, University of Crete, P.O. Box 2208, GR-710 03 Heraklion, Crete, Greece – and – Institute of Applied and Computational Mathematics, FORTH, P.O. Box 1527, GR-711 10 Heraklion, Crete, Greece
Email: gkarali@tem.uoc.gr

M. Plexousakis
Affiliation: Department of Mathematics and Applied Mathematics, University of Crete, P.O. Box 2208, GR-710 03 Heraklion, Crete, Greece – and – Institute of Applied and Computational Mathematics, FORTH, P.O. Box 1527, GR-711 10 Heraklion, Crete, Greece
Email: plex@tem.uoc.gr

G. E. Zouraris
Affiliation: Department of Mathematics and Applied Mathematics, University of Crete, P.O. Box 2208, GR-710 03 Heraklion, Crete, Greece – and – Institute of Applied and Computational Mathematics, FORTH, P.O. Box 1527, GR-711 10 Heraklion, Crete, Greece
Email: zouraris@math.uoc.gr

DOI: https://doi.org/10.1090/S0025-5718-2014-02900-1
Keywords: Schr{\"o}dinger-type equation, noncylindrical domain, Robin-type boundary condition, elliptic regularity, Crank-Nicolson time stepping, finite element method, apriori error estimates, underwater acoustics.
Received by editor(s): September 1, 2011
Received by editor(s) in revised form: October 29, 2012, and October 3, 2013
Published electronically: November 5, 2014
Dedicated: Dedicated to Professor Vassilios Dougalis on the occasion of his 65th birthday
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.