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

TL;DR

This paper establishes new uniform error estimates for a Crank-Nicolson-polylinear finite element method with discrete transparent boundary conditions applied to the generalized Schrödinger equation in unbounded parallelepipeds, demonstrating stability and superconvergence.

Contribution

It provides the first proof of uniform error estimates for this numerical scheme with discrete TBCs for the generalized Schrödinger equation.

Findings

Error estimates of order O(τ^2 + |h|^2) in L^2 norm

Superconvergence in H^1 norm for dimensions 1 to 3

Stability properties of the method

Abstract

We deal with an initial-boundary value problem for the generalized time-dependent Schr\"odinger equation with variable coefficients in an unbounded $n$--dimensional parallelepiped ($n\geq 1$). 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 $O(\tau^2+|h|^2)$ uniformly in time in $L^2$ space norm, for $n\geq 1$, and mesh $H^1$ space norm, for $1\leq n\leq 3$ (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.