Discontinuous Galerkin and multiscale variational schemes for a coupled damped nonlinear system of Schrödinger equations (Q2864597)

The authors construct a streamline diffusion-based discontinuous Galerkin scheme for solving a coupled system of nonlinear Schrödinger equations and extend the resulting method to a multiscale variational scheme. Stability estimates are proved and optimal convergence rates are obtained. In the weak formulation, to make the underlying bilinear form coercive, it is necessary to supply the equation system with an artificial viscosity term with a small coefficient of order proportional to a power of mesh size. The original and multi-schemes are numerically tested by implementing an example of an application of the time-dependent Schrödinger equation to the coupled ultrafast laser beam.