A Space-Time Trefftz Discontinuous Galerkin Method for the Linear Schrödinger Equation

A space-time Trefftz discontinuous Galerkin method for the Schr\"odinger equation with piecewise-constant potential is proposed and analyzed. Following the spirit of Trefftz methods, trial and test spaces are spanned by nonpolynomial complex wave functions that satisfy the Schr\"odinger equation locally on each element of the space-time mesh. This allows for a significant reduction in the number of degrees of freedom in comparison with full polynomial spaces. We prove well-posedness and stability of the method and, for the one- and two-dimensional cases, optimal, high-order, h-convergence error estimates in a skeleton norm. Some numerical experiments validate the theoretical results presented.