Analysis of an implicit fully discrete local discontinuous Galerkin method for the time-fractional Schrödinger equation

• Published in: Finite elements in analysis and design Vol. 59; pp. 28 - 34
• Main Authors: Wei, Leilei, He, Yinnian, Zhang, Xindong, Wang, Shaoli
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.10.2012; Elsevier
• Subjects: Error estimates; Exact sciences and technology; Local discontinuous Galerkin method; Mathematical analysis; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Ordinary differential equations; Partial differential equations, initial value problems and time-dependant initial-boundary value problems; Physics; Quantum statistical mechanics; Schrödinger equation; Sciences and techniques of general use; Stability; Statistical physics, thermodynamics, and nonlinear dynamical systems; Time-fractional partial differential equations; Time-fractional partial differential equations; Error estimates; Stability; Schrödinger equation; Local discontinuous Galerkin method; Discontinuity; Error analysis; Error estimation; Modeling; Partial differential equation; Fractional derivative; Finite element method; Convergence rate; Galerkin method; Finite difference method
• ISSN: 0168-874X; 1872-6925
• DOI: 10.1016/j.finel.2012.03.008

In this paper we present and analyze an implicit fully discrete local discontinuous Galerkin (LDG) finite element method for solving the time-fractional Schrödinger equation, where the fractional derivative is described in the Caputo sense. The scheme is based on a finite difference method in time and local discontinuous Galerkin methods in space. A stability and error analysis is performed on the numerical methods. Numerical results confirm the expected convergence rates and illustrate the effectiveness of the method. ► An implicit fully discrete LDG method is presented. ► The L2 error estimate is new. ► An unconditional stable finite element method for solving time-fractional Schrödinger equation.