A Spacetime DPG Method for the Schrödinger Equation

A spacetime discontinuous Petrov--Galerkin (DPG) method for the linear time-dependent Schrodinger equation is proposed. The spacetime approach is particularly attractive for capturing irregular solutions. Motivated by the fact that some irregular Schrodinger solutions cannot be solutions of certain first order reformulations, the proposed spacetime method uses the second order Schrodinger operator. Two variational formulations are proved to be well posed: a strong formulation (with no relaxation of the original equation) and a weak formulation (also called the “ultraweak formulation,” which transfers all derivatives onto test functions). The convergence of the DPG method based on the ultraweak formulation is investigated using an interpolation operator. A stand-alone appendix analyzes the ultraweak formulation for general differential operators. Reports of numerical experiments motivated by pulse propagation in dispersive optical fibers are also included.