Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

Request Permissions   Purchase Content 
 
 

 

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [1] Leif Abrahamsson and Heinz-Otto Kreiss, The initial-boundary value problem for the Schrödinger equation, Math. Methods Appl. Sci. 13 (1990), no. 5, 385-390. MR 1078588 (91h:35093), https://doi.org/10.1002/mma.1670130503
  • [2] Leif Abrahamsson and Heinz-Otto Kreiss, Boundary conditions for the parabolic equation in a range-dependent duct, J. Acoust. Soc. Amer. 87 (1990), no. 6, 2438-2441. MR 1054995 (91b:76090), https://doi.org/10.1121/1.399089
  • [3] Georgios D. Akrivis and Vassilios A. Dougalis, Finite difference discretization with variable mesh of the Schrödinger equation in a variable domain, Bull. Soc. Math. Grèce (N.S.) 31 (1990), 19-28. MR 1108904 (93a:65111)
  • [4] G. D. Akrivis, V. A. Dougalis, and G. E. Zouraris, Error estimates for finite difference methods for a wide-angle ``parabolic'' equation, SIAM J. Numer. Anal. 33 (1996), no. 6, 2488-2509. MR 1427476 (98a:65110), https://doi.org/10.1137/S0036142994266352
  • [5] G. D. Akrivis, V. A. Dougalis, and G. E. Zouraris, Finite difference schemes for the ``parabolic'' equation in a variable depth environment with a rigid bottom boundary condition, SIAM J. Numer. Anal. 39 (2001), no. 2, 539-565 (electronic). MR 1860264 (2002g:65090), https://doi.org/10.1137/S0036142999367460
  • [6] D. C. Antonopoulou, Theory and numerical analysis of parabolic approximations, Ph.D. Thesis, University of Athens, Greece, 2006 (in Greek).
  • [7] D. C. Antonopoulou, V. A. Dougalis, F. Sturm and G. E. Zouraris, Conservative initial-boundary value problems for the wide-angle PE in waveguides with variable bottoms, Proceedings of the 9th European Conference on Underwater Acoustics (9th EQUA), M. E. Zakharia, D. Cassereau and F. Luppé, eds. 1, 375-380 (2008).
  • [8] D. C. Antonopoulou, V. A. Dougalis, and G. E. Zouraris, Galerkin methods for parabolic and Schrödinger equations with dynamical boundary conditions and applications to underwater acoustics, SIAM J. Numer. Anal. 47 (2009), no. 4, 2752-2781. MR 2551145 (2010m:65212), https://doi.org/10.1137/070710858
  • [9] Dimitra C. Antonopoulou, Vassilios A. Dougalis, and Georgios E. Zouraris, A finite difference method for the wide-angle ``parabolic'' equation in a waveguide with downsloping bottom, Numer. Methods Partial Differential Equations 29 (2013), no. 4, 1416-1440. MR 3053872, https://doi.org/10.1002/num.21762
  • [10] D. C. Antonopoulou and M. Plexousakis, Discontinuous Galerkin methods for the linear Schrödinger equation in non-cylindrical domains, Numer. Math. 115 (2010), no. 4, 585-608. MR 2658156 (2011e:65184), https://doi.org/10.1007/s00211-010-0296-5
  • [11] A. Bamberger, B. Engquist, L. Halpern, and P. Joly, Parabolic wave equation approximations in heterogenous media, SIAM J. Appl. Math. 48 (1988), no. 1, 99-128. MR 923293 (89h:35225), https://doi.org/10.1137/0148005
  • [12] Susanne C. Brenner and L. Ridgway Scott, The Mathematical Theory of Finite Element Methods, 3rd ed., Texts in Applied Mathematics, vol. 15, Springer, New York, 2008. MR 2373954 (2008m:65001)
  • [13] V. A. Dougalis, F. Sturm, and G. E. Zouraris, On an initial-boundary value problem for a wide-angle parabolic equation in a waveguide with a variable bottom, Math. Methods Appl. Sci. 32 (2009), no. 12, 1519-1540. MR 2535860 (2011a:35212), https://doi.org/10.1002/mma.1097
  • [14] Todd Dupont, $ L^{2}$-estimates for Galerkin methods for second order hyperbolic equations, SIAM J. Numer. Anal. 10 (1973), 880-889. MR 0349045 (50 #1539)
  • [15] L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, 1998. MR 1625845
  • [16] Fritz John, Partial Differential Equations, 4th ed., Applied Mathematical Sciences, vol. 1, Springer-Verlag, New York, 1982. MR 831655 (87g:35002)
  • [17] J. L. Lions and E. Magénes, Problèmes aux Limites Non Homogènes et Applications, I, Dunod, Paris, 1968. MR 0247243
  • [18] F. Sturm, Modélisation mathématique et numérique d' un problème de propagation en acoustique sous-marine: prise en compte d'un environnement variable tridimensionnel, Thèse de Docteur en Sciences Université de Toulon et du Var, France, 1997.
  • [19] Fred D. Tappert, The parabolic approximation method, Wave propagation and underwater acoustics (Workshop, Mystic, Conn., 1974), Springer, Berlin, 1977, pp. 224-287. Lecture Notes in Phys., Vol. 70. MR 0475274 (57 #14891)
  • [20] Vidar Thomée, Galerkin Finite Element Methods for Parabolic Problems, Springer Series in Computational Mathematics, vol. 25, Springer-Verlag, Berlin, 1997. MR 1479170 (98m:65007)
  • [21] Juan Luis Vázquez and Enzo Vitillaro, Heat equation with dynamical boundary conditions of reactive type, Comm. Partial Differential Equations 33 (2008), no. 4-6, 561-612. MR 2424369 (2009h:35180), https://doi.org/10.1080/03605300801970960
  • [22] G. E. Zouraris, Analysis of numerical methods for evolution partial differential equations, Ph.D. Thesis, University of Crete, Greece, 1995 (in Greek).

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 65M12, 65M15, 65M60

Retrieve articles in all journals with MSC (2000): 65M12, 65M15, 65M60


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.

American Mathematical Society