An alternating direction implicit finite element Galerkin method for the linear Schrödinger equation

We formulate and analyze a fully discrete approximate solution of the linear Schrödinger equation on the unit square written as a Schrödinger-type system. The finite element Galerkin method is used for the spatial discretization, and the time stepping is done with an alternating direction implicit extrapolated Crank-Nicolson method. We demonstrate the existence and uniqueness of the approximation, and prove that the scheme is of optimal accuracy in the \(L^2, H^1\) $L^2,H^1$ and \(L^{\infty }\) $L^{\infty }$ norms in space and second-order accurate in time. Numerical results are presented which support the theory.