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/
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