On accuracy of the mass-preserving DG method to multi-dimensional Schrodinger equations

In this paper, we present two approaches to the error analysis of a semidiscrete mass-preserving discontinuous Galerkin method, introduced by Lu, Huang and Liu (2015, Mass preserving direct discontinuous Galerkin methods for Schrodinger equations. J. Comp. Phys., 282, 210-226), for the solution of multi-dimensional Schrodinger equations. The first approach is based on an explicit global projection using tensor product polynomials on rectangular meshes. The L-2 error bound obtained is optimal, independent of the size of the flux parameter. The second approach is based on an implicit global projection using standard polynomials on arbitrary shape-regular meshes. The L-2 error bound obtained for this method is also optimal, but it is valid only when the flux parameter is sufficiently large. Numerical experiments are presented to demonstrate the theoretical results.