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

TY - JOUR

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 Schrodinger-type equations in the L8(L(2)- 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 et al. in [Math. Comput. 75 (2006) 511-531], leads to a posteriori upper bounds that are of optimal order in the L8(L2)-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 Schrodinger-type equations in the L8(L(2)- 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 et al. in [Math. Comput. 75 (2006) 511-531], leads to a posteriori upper bounds that are of optimal order in the L8(L2)-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 - TIME

KW - energy techniques

KW - GALERKIN METHOD

KW - a posteriori error analysis

KW - Crank-Nicolson method

KW - Linear Schrodinger equation

KW - FINITE-ELEMENT-METHOD

KW - L(infinity)(L(2))- and L(infinity)(H(1))-norm

KW - Crank-Nicolson reconstruction

U2 - 10.1051/m2an/2010101

DO - 10.1051/m2an/2010101

M3 - Article

SN - 0764-583X

VL - 45

SP - 761

EP - 778

JO - Esaim: Mathematical Modelling and Numerical Analysis

JF - Esaim: Mathematical Modelling and Numerical Analysis

IS - 4

ER -