TY - JOUR
T1 - A Posteriori Error Analysis for Evolution Nonlinear Schrödinger Equations up to the Critical Exponent
AU - Katsaounis, Theodoros
AU - Kyza, Irene
N1 - KAUST Repository Item: Exported on 2021-02-23
Acknowledgements: The work of the authors was partially supported by Excellence Award 1456 of the Greek Ministry of Research and Education. The second author is grateful to Prof. Charalambos Makridakis for suggesting the problem and for fruitful discussions. The authors would like to thank Prof. Georgios Akrivis and the anonymous reviewers for their valuable comments and suggestions.
PY - 2018/5/17
Y1 - 2018/5/17
N2 - We provide a posteriori error estimates in the L([0, T]; L(?))-norm for relaxation time discrete and fully discrete schemes for a class of evolution nonlinear Schrödinger equations up to the critical exponent. In particular for the discretization in time we use the relaxation Crank–Nicolson-type scheme introduced by Besse in [SIAM J. Numer. Anal., 42 (2004), pp. 934–952]. The space discretization consists of finite element spaces that are allowed to change between time steps. The estimates are obtained using the reconstruction technique. Through this technique the problem is converted to a perturbation of the original partial differential equation and this makes it possible to use nonlinear stability arguments as in the continuous problem. Our analysis includes as special cases the cubic and quintic nonlinear Schrödinger equations in one spatial dimension and the cubic nonlinear Schrödinger equation in two spatial dimensions. Numerical results illustrate that the estimates are indeed of optimal order of convergence.
AB - We provide a posteriori error estimates in the L([0, T]; L(?))-norm for relaxation time discrete and fully discrete schemes for a class of evolution nonlinear Schrödinger equations up to the critical exponent. In particular for the discretization in time we use the relaxation Crank–Nicolson-type scheme introduced by Besse in [SIAM J. Numer. Anal., 42 (2004), pp. 934–952]. The space discretization consists of finite element spaces that are allowed to change between time steps. The estimates are obtained using the reconstruction technique. Through this technique the problem is converted to a perturbation of the original partial differential equation and this makes it possible to use nonlinear stability arguments as in the continuous problem. Our analysis includes as special cases the cubic and quintic nonlinear Schrödinger equations in one spatial dimension and the cubic nonlinear Schrödinger equation in two spatial dimensions. Numerical results illustrate that the estimates are indeed of optimal order of convergence.
UR - http://hdl.handle.net/10754/628360
UR - https://epubs.siam.org/doi/10.1137/16M1108029
UR - http://www.scopus.com/inward/record.url?scp=85049429191&partnerID=8YFLogxK
U2 - 10.1137/16M1108029
DO - 10.1137/16M1108029
M3 - Article
AN - SCOPUS:85049429191
VL - 56
SP - 1405
EP - 1434
JO - SIAM Journal on Numerical Analysis
JF - SIAM Journal on Numerical Analysis
SN - 0036-1429
IS - 3
ER -