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A posteriori error analysis for the Crank-Nicolson method for linear Schrödinger equations. (English) Zbl 1269.65088

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.

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

Citations:

Zbl 1101.65094
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