A posteriori error analysis for evolution nonlinear schrödinger equations up to the critical exponent

Theodoros Katsaounis; Irene Kyza

doi:10.1137/16M1108029

A posteriori error analysis for evolution nonlinear schrödinger equations up to the critical exponent

Theodoros Katsaounis, Irene Kyza

Computer, Electrical and Mathematical Sciences and Engineering

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

We provide a posteriori error estimates in the L⁸([0, T]; L²(?))-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.

Original language	English (US)
Pages (from-to)	1405-1434
Number of pages	30
Journal	SIAM Journal on Numerical Analysis
Volume	56
Issue number	3
DOIs	https://doi.org/10.1137/16M1108029
State	Published - 2018

Bibliographical note

Funding Information:
∗Received by the editors December 14, 2016; accepted for publication (in revised form) March 5, 2018; published electronically May 17, 2018. http://www.siam.org/journals/sinum/56-3/M110802.html Funding: The work of the authors was partially supported by Excellence Award 1456 of the Greek Ministry of Research and Education. †Computer, Electrical and Mathematical Sciences & Engineering, KAUST, Saudi Arabia, IACM– FORTH, Heraklion, Greece, and Department of Mathematics and Applied Mathematics, University of Crete, Greece (theodoros.katsaounis@kaust.edu.sa). ‡Department of Mathematics, University of Dundee, Dundee DD1 4HN, Scotland, UK (ikyza@ dundee.ac.uk).

Funding Information:
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.

Publisher Copyright:
© 2018 Society for Industrial and Applied Mathematics.

Keywords

A posteriori error control
Evolution NLS
Finite elements
Power nonlinearities
Reconstruction technique
Relaxation Crank–Nicolson-type scheme

ASJC Scopus subject areas

Numerical Analysis
Computational Mathematics
Applied Mathematics

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10.1137/16M1108029

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title = "A posteriori error analysis for evolution nonlinear schr{\"o}dinger equations up to the critical exponent",

abstract = "We provide a posteriori error estimates in the L8([0, T]; L2(?))-norm for relaxation time discrete and fully discrete schemes for a class of evolution nonlinear Schr{\"o}dinger 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 Schr{\"o}dinger equations in one spatial dimension and the cubic nonlinear Schr{\"o}dinger equation in two spatial dimensions. Numerical results illustrate that the estimates are indeed of optimal order of convergence.",

keywords = "A posteriori error control, Evolution NLS, Finite elements, Power nonlinearities, Reconstruction technique, Relaxation Crank–Nicolson-type scheme",

author = "Theodoros Katsaounis and Irene Kyza",

note = "Funding Information: ∗Received by the editors December 14, 2016; accepted for publication (in revised form) March 5, 2018; published electronically May 17, 2018. http://www.siam.org/journals/sinum/56-3/M110802.html Funding: The work of the authors was partially supported by Excellence Award 1456 of the Greek Ministry of Research and Education. †Computer, Electrical and Mathematical Sciences & Engineering, KAUST, Saudi Arabia, IACM– FORTH, Heraklion, Greece, and Department of Mathematics and Applied Mathematics, University of Crete, Greece (theodoros.katsaounis@kaust.edu.sa). ‡Department of Mathematics, University of Dundee, Dundee DD1 4HN, Scotland, UK (ikyza@ dundee.ac.uk). Funding Information: 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. Publisher Copyright: {\textcopyright} 2018 Society for Industrial and Applied Mathematics.",

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doi = "10.1137/16M1108029",

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AU - Kyza, Irene

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PY - 2018

Y1 - 2018

N2 - We provide a posteriori error estimates in the L8([0, T]; L2(?))-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 L8([0, T]; L2(?))-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.

KW - A posteriori error control

KW - Evolution NLS

KW - Finite elements

KW - Power nonlinearities

KW - Reconstruction technique

KW - Relaxation Crank–Nicolson-type scheme

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EP - 1434

JO - SIAM Journal on Numerical Analysis

JF - SIAM Journal on Numerical Analysis

SN - 0036-1429

IS - 3

ER -

A posteriori error analysis for evolution nonlinear schrödinger equations up to the critical exponent

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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