An adaptive nonconforming finite element method for the nonlinear Schrödinger equation

A fully discrete approximation to the solution of the two-dimensional Nonlinear Schrödinger Equation is presented and analyzed. The spatial discretization is based on non-conforming finite elements and the temporal discretization is achieved by a variable step Crank-Nicolson scheme. The numerical method presented here is aimed at approximating general (i.e. not necessarily radially symmetric solutions) of the Nonlinear Schrodinger equation which emanate from general initial profiles (vanishing at infinity). In particular, we apply it to study the formation of singularities. The numerical scheme employs a spatial refinement strategy which allows the mesh to be refined locally and without compromising the overall quality of the mesh. The adaptive refinement strategy is effected by a simple and computationaly efficient test, based on inverse estimates on the approximating subspace. Several numerical experiments concerning the formation of singularities are presented.