In this paper, we consider a two-dimensional Schrödinger-type equation with a dynamical boundary condition. This model describes the long-range sound propagation in naval environments of variable rigid bottom topography. Our choice for a regular enough finite element approximation is motivated by the dynamical condition and therefore, consists of a cubic splines implicit Galerkin method in space. Furthermore, we apply a Crank–Nicolson time stepping for the evolutionary variable. We prove existence and stability of the semidiscrete and fully discrete solution. Due to the complexity of the analyzed problem, we use very refined technics in order to derive estimates of the numerical error in the -norm.
DOI : 10.1051/m2an/2015004
Mots clés : 2-D Schrödinger equation, finite element methods, error estimates, noncylindrical domain, Neumann boundary condition, cubic splines, Crank–Nicolson time stepping, dynamical boundary condition, underwater acoustics
@article{M2AN_2015__49_4_1127_0,
author = {Antonopoulou, D. C.},
title = {Galerkin methods for a {Schr\"odinger-type} equation with a dynamical boundary condition in two dimensions},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1127--1156},
publisher = {EDP-Sciences},
volume = {49},
number = {4},
year = {2015},
doi = {10.1051/m2an/2015004},
mrnumber = {3371906},
zbl = {1327.65190},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2015004/}
}
TY - JOUR AU - Antonopoulou, D. C. TI - Galerkin methods for a Schrödinger-type equation with a dynamical boundary condition in two dimensions JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2015 SP - 1127 EP - 1156 VL - 49 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2015004/ DO - 10.1051/m2an/2015004 LA - en ID - M2AN_2015__49_4_1127_0 ER -
%0 Journal Article %A Antonopoulou, D. C. %T Galerkin methods for a Schrödinger-type equation with a dynamical boundary condition in two dimensions %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2015 %P 1127-1156 %V 49 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2015004/ %R 10.1051/m2an/2015004 %G en %F M2AN_2015__49_4_1127_0
Antonopoulou, D. C. Galerkin methods for a Schrödinger-type equation with a dynamical boundary condition in two dimensions. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 4, pp. 1127-1156. doi : 10.1051/m2an/2015004. http://www.numdam.org/articles/10.1051/m2an/2015004/
and , The initial boundary value problem for the Schrödinger equation. Math. Methods Appl. Sci. 13 (1990a) 385–390. | DOI | MR | Zbl
and , Boundary conditions for the parabolic equation in a range-dependent duct. J. Acoust. Soc. Amer. 87 (1990b) 2438–2441. | DOI | MR
and , Finite difference discretization with variable mesh of the Schrödinger equation in a variable domain. Bull. Greek Math. Soc. 31 (1990) 19–28. | MR | Zbl
, and , Finite difference schemes for the ‘Parabolic’ Equation in a variable depth environment with a rigid bottom boundary condition. SIAM J. Numer. Anal. 39 (2001) 539-565. | DOI | MR | Zbl
D.C. Antonopoulou, Theory and numerical analysis of parabolic approximations. Ph.D. thesis (in Greek). University of Athens, Greece (2006).
, Galerkin methods for the ‘Parabolic Equation’ Dirichlet problem in a variable 2-D and 3-D topography, Appl. Numer. Math. 67 (2013) 17–34. | DOI | MR | Zbl
and , Discontinuous Galerkin methods for the linear Schrödinger equation in noncylindrical domains. Numer. Math. 115 (2010) 585–608. | DOI | MR | Zbl
, and , Galerkin methods for parabolic and Schrödinger equations with dynamical boundary conditions and applications to underwater acoustics. SIAM J. Numer. Anal. 47 (2009) 2752–2781. | DOI | MR | Zbl
, , and , Crank–Nicolson finite element discretizations for a two dimensional linear Schrödinger-type equation posed in a noncylindrical domain. Math. Comput. 84 (2015) 1571–1598. | DOI | MR | Zbl
S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. In vol. 15 of Texts Appl. Math. Springer-Verlag, New York (1994). | MR | Zbl
V.A. Dougalis, N.A. Kampanis, F. Sturm, and G.E. Zouraris, Numerical solution of the Parabolic Equation in range-dependent waveguides in Effective Computational Methods for Wave Propagation. Edited by N.A. Kampanis et al. Chapman and Hall/CRC, Boca Raton (2008) 175–207. | MR | Zbl
M. Ehrhardt, Discrete artificial boundary conditions. Ph.D. thesis, Technische Universität, Berlin (2001).
, Quasilinear parabolic systems with dynamical boundary conditions, Commun. Partial. Differ. Eqs. 18 (1993) 1309–1364. | DOI | MR | Zbl
L.C. Evans, Partial Differential Equations. In vol. 19 of Grad. Stud. Math. American Mathematical Society (1998). | MR | Zbl
F.B. Jensen, W.A. Kuperman, M.B. Porter, H. Schmidt, Computational Ocean Acoustics. AIP Series in Modern Acoustics and Signal Processing. American Institute of Physics, New York (1994). | MR | Zbl
M.H. Schultz, Spline Analysis. Prentice-Hall (1973). | MR | Zbl
F. Sturm, Modélisation mathématique et numérique d’ un problème de propagation en acoustique sous-marine: prise en compte d’un environnement variable tridimensionnel. Ph.D. thesis, Université de Toulon et du Var, France (1997).
F.D. Tappert, The parabolic approximation method, in Wave Propagation and Underwater Acoustics, edited by J.B. Keller and J.S. Papadakis. In vol. 70 of Lect. Notes Phys. Springer-Verlag, Berlin (1977) 224-287. | MR
and , Heat equation with dynamical boundary conditions of reactive type. Commun. Partial. Differ. Eqs. 33 (2008) 561–612. | DOI | MR | Zbl
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