Kyza, Irene A posteriori error analysis for the Crank-Nicolson method for linear Schrödinger equations. (English) Zbl 1269.65088 ESAIM, Math. Model. Numer. Anal. 45, No. 4, 761-778 (2011). Summary: We prove a posteriori error estimates of optimal order for linear Schrödinger-type equations in the \(L{^\infty}(L{^2})\)- and the \(L{^\infty}(H{^1})\)-norm. We discretize only in time by the Crank-Nicolson method. The direct use of the reconstruction technique, as it has been proposed by G. Akrivis et al. [Math. Comput. 75, No. 254, 511–531 (2006; Zbl 1101.65094)], leads to a posteriori upper bounds that are of optimal order in the \(L{^\infty}(L{^2})\)-norm, but of suboptimal order in the \(L{^\infty}(H{^1})\)-norm. The optimality in the case of \(L{^\infty}(H{^1})\)-norm is recovered by using an auxiliary initial- and boundary-value problem. Cited in 5 Documents MSC: 65M15 Error bounds for initial value and initial-boundary value problems involving PDEs 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 35Q41 Time-dependent Schrödinger equations and Dirac equations Keywords:linear Schrödinger equation; Crank-Nicolson method; Crank-Nicolson reconstruction; a posteriori error analysis; energy techniques; \(L{^\infty }(L^{2})\)- and \(L{^\infty }(H^{1})\)-norm; numerical examples Citations:Zbl 1101.65094 PDFBibTeX XMLCite \textit{I. Kyza}, ESAIM, Math. Model. Numer. Anal. 45, No. 4, 761--778 (2011; Zbl 1269.65088) Full Text: DOI