Uniform in time error analysis of HDG approximation for schrödinger equation based on HDG projection

This paper presents error analysis of hybridizable discontinuous Galerkin (HDG) timedomain method for solving time dependent Schrödinger equations. The numerical trace and numerical ux are constructed to preserve the conservative property for the density of the particle described. We prove that there exist the superconvergence properties of the HDG method, which do hold for second-order elliptic problems, uniformly in time for the semidiscretization by the same method of Schrödinger equations provided that enough regularity is satisfied. Thus, if the approximations are piecewise polynomials of degree r, the approximations to the wave function and the ux converge with order r + 1. The suitably chosen projection of the wave function into a space of lower polynomial degree superconverges with order r + 2 for r ≥ 1 uniformly in time. The application of element-byelement postprocessing of the approximate solution which provides an approximation of the potential convergence with order r + 2 for r ≥ 1 in L² is also uniformly in time.