A time- and spaceadaptive algorithm for the linear time-dependent Schrödinger equation

Summary.

We prove an a posteriori error estimate for the linear time-dependent Schrödinger equation in ${\Bbb R}^N$. From this, we derive a residual based local error estimator that allows us to adjust the mesh and the time step size in order to obtain a numerical solution with a prescribed accuracy. As a special feature, the error estimator controls localization and size of the finite computational domain in each time step. An algorithm is described to compute this solution and numerical results in one space dimension are included.