On Error Estimates of the Crank-Nicolson-Polylinear Finite Element Method with the Discrete TBC for the Generalized Schrödinger Equation in an Unbounded Parallelepiped

2015/03/26 by Zlotnik Alexander, Zlotnik, Alexander
Engineering · Mathematics · #35Q40 #65M12 #65M60 #Differential Equations and Numerical Methods #Electromagnetic Simulation and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods for differential equations

paper · doi:10.48550/arxiv.1503.07811

openalex publication_date 2015/03/26 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

We deal with an initial-boundary value problem for the generalized time-dependent Schrödinger equation with variable coefficients in an unbounded n--dimensional parallelepiped (n≥ 1). To solve it, the Crank-Nicolson in time and the polylinear finite element in space method with the discrete transpa\-rent boundary conditions is considered. We present its stability properties and derive new error estimates O(τ²+|h|²) uniformly in time in L² space norm, for n≥ 1, and mesh H¹ space norm, for 1≤ n≤ 3 (a superconvergence result), under the Sobolev-type assumptions on the initial function. Such estimates are proved for methods with the discrete TBCs for the first time.

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