Error Estimates of the Crank-Nicolson-Polylinear FEM with the Discrete TBC for the Generalized Schrodinger Equation in an Unbounded Parallelepiped

摘要

We deal with an initial-boundary value problem for the generalized time-dependent Schrodinger 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 transparent boundary conditions is considered. We present its stability properties and derive new error estimates O(τ~2 + |h|~2) uniformly in time in L~2 space norm, for n ≥ 1, and mesh H~1 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.