Convergence of a finite element scheme for the two-dimensional time-dependent Schrödinger equation in a long strip (Q966091)

The authors construct and analyze a fully discrete finite element scheme for a time-dependent Schrödinger equation in an infinitely long channel. The problem is reduced to an initial-boundary value problem on a bounded domain by introducing transparent artificial boundaries. The reduced problem is discretized and a fully discrete scheme is constructed based on a bilinear or quadratic finite element approximation in space and the Crank-Nicolson scheme in time. A rigorous analysis proves the unconditional stability of the scheme and its convergence. The order of convergence is also obtained. Two numerical examples are given to verify the accuracy of the scheme.