A posteriori error analysis for the Crank-Nicolson method for linear Schrödinger equations

Irene Kyza

doi:10.1051/m2an/2010101

A posteriori error analysis for the Crank-Nicolson method for linear Schrödinger equations

Irene Kyza^*

^*Corresponding author for this work

Applied Mathematics

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

Abstract

We prove a posteriori error estimates of optimal order for linear Schrödinger-type equations in the L⁸(L²)- and the L⁸(H¹)-norm. We discretize only in time by the Crank-Nicolson method. The direct use of the reconstruction technique, as it has been proposed by Akrivis etal. in [Math. Comput. 75 (2006) 511531], leads to a posteriori upper bounds that are of optimal order in the L⁸(L ²)-norm, but of suboptimal order in the L⁸(H ¹)-norm. The optimality in the case of L⁸(H ¹)-norm is recovered by using an auxiliary initial- and boundary-value problem.

Original language	English
Pages (from-to)	761-778
Number of pages	18
Journal	ESAIM: Mathematical Modelling and Numerical Analysis
Volume	45
Issue number	4
DOIs	https://doi.org/10.1051/m2an/2010101
Publication status	Published - Jul 2011

Keywords

A posteriori error analysis
Crank-Nicolson method
Crank-Nicolson reconstruction
Energy techniques
L=(L2)- and L=(H1)-norm
Linear Schrödinger equation

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title = "A posteriori error analysis for the Crank-Nicolson method for linear Schr{\"o}dinger equations",

abstract = "We prove a posteriori error estimates of optimal order for linear Schr{\"o}dinger-type equations in the L8(L2)- and the L8(H1)-norm. We discretize only in time by the Crank-Nicolson method. The direct use of the reconstruction technique, as it has been proposed by Akrivis etal. in [Math. Comput. 75 (2006) 511531], leads to a posteriori upper bounds that are of optimal order in the L8(L 2)-norm, but of suboptimal order in the L8(H 1)-norm. The optimality in the case of L8(H 1)-norm is recovered by using an auxiliary initial- and boundary-value problem.",

keywords = "A posteriori error analysis, Crank-Nicolson method, Crank-Nicolson reconstruction, Energy techniques, L=(L2)- and L=(H1)-norm, Linear Schr{\"o}dinger equation",

author = "Irene Kyza",

year = "2011",

month = jul,

doi = "10.1051/m2an/2010101",

language = "English",

volume = "45",

pages = "761--778",

journal = "ESAIM: Mathematical Modelling and Numerical Analysis",

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T1 - A posteriori error analysis for the Crank-Nicolson method for linear Schrödinger equations

AU - Kyza, Irene

PY - 2011/7

Y1 - 2011/7

N2 - We prove a posteriori error estimates of optimal order for linear Schrödinger-type equations in the L8(L2)- and the L8(H1)-norm. We discretize only in time by the Crank-Nicolson method. The direct use of the reconstruction technique, as it has been proposed by Akrivis etal. in [Math. Comput. 75 (2006) 511531], leads to a posteriori upper bounds that are of optimal order in the L8(L 2)-norm, but of suboptimal order in the L8(H 1)-norm. The optimality in the case of L8(H 1)-norm is recovered by using an auxiliary initial- and boundary-value problem.

AB - We prove a posteriori error estimates of optimal order for linear Schrödinger-type equations in the L8(L2)- and the L8(H1)-norm. We discretize only in time by the Crank-Nicolson method. The direct use of the reconstruction technique, as it has been proposed by Akrivis etal. in [Math. Comput. 75 (2006) 511531], leads to a posteriori upper bounds that are of optimal order in the L8(L 2)-norm, but of suboptimal order in the L8(H 1)-norm. The optimality in the case of L8(H 1)-norm is recovered by using an auxiliary initial- and boundary-value problem.

KW - A posteriori error analysis

KW - Crank-Nicolson method

KW - Crank-Nicolson reconstruction

KW - Energy techniques

KW - L=(L2)- and L=(H1)-norm

KW - Linear Schrödinger equation

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DO - 10.1051/m2an/2010101

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VL - 45

SP - 761

EP - 778

JO - ESAIM: Mathematical Modelling and Numerical Analysis

JF - ESAIM: Mathematical Modelling and Numerical Analysis

IS - 4

ER -

A posteriori error analysis for the Crank-Nicolson method for linear Schrödinger equations

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