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.

Zlotnik Alexander

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.

Keywords: метод конечных элементов, устойчивость, неограниченная область, discrete transparent boundary conditions, stability, дискретные прозрачные граничные условия, The time-dependent Schrödinger equation, finite element method, unbounded domains, нестационарное уравнение Шрёдингера, the Crank-Nicolson discretization in time, дискретизация Кранк-Никольсон по времени, оценки погрешности, error estimates, superconvergence, суперсходимость

Publication based on the results of: Построение и анализ численных методов решения некоторых задач для эволюционных уравнений математической физики (2014)