Superconvergence of UltraWeak Discontinuous Galerkin Methods for the Linear Schrxc3xb6dinger Equation in One Dimension We analyze the superconvergence properties of ultraweak discontinuous Galerkin (UWDG) methods with various choices of flux parameters for onedimensional linear Schrodinger equation. In our previous work (Chen et al. in J Sci Comput 78(2):772xe2x80x93815, 2019), stability and optimal convergence rate are established for a large class of flux parameters. Depending on the flux choices and if the polynomial degree k is even or odd, in this paper, we prove 2k or (2k1)th order superconvergence rate for cell averages and numerical flux of the function, as well as (2k1) or (2k2)th order for numerical flux of the derivative. In addition, we prove superconvergence of (k2) or (k3)th order of the DG solution towards a special projection. At a class of special points, the function values and the first and second order derivatives of the DG solution are superconvergent with order k2, k1, k, respectively. The proof relies on the correction function techniques initiated in Cao et al. (SIAM J Numer Anal 52(5):2555xe2x80x932573, 2014), and applied to Cao et al. (Numer Methods Partial Differ Equ 33(1):290xe2x80x93317, 2017) for direct DG (DDG) methods for diffusion problems. Compared with Cao et al. (2017), Schrodinger equation poses unique challenges for superconvergence proof because of the lack of the dissipation mechanism from the equation. One major highlight of our proof is that we introduce specially chosen test functions in the error equation and show the superconvergence of the second derivative and jump across the cell interfaces of the difference between numerical solution and projected exact solution. This technique was originally proposed in Cheng and Shu (SIAM J Numer Anal 47(6):4044xe2x80x934072, 2010) and is essential to elevate the convergence order for our analysis. Finally, by negative norm estimates, we apply the postprocessing technique and show that the accuracy of our scheme can be enhanced to order 2k. Theoretical results are verified by numerical experiments.

Superconvergent Functional Estimates from TensorProduct Generalized SummationbyParts Discretizations in Curvilinear Coordinates We investigate superconvergent functional estimates in curvilinear coordinates for diagonalnorm tensorproduct generalized summationbyparts operators. We show that interpolationextrapolation operators of degree greater than or equal to 2p are required to preserve at least 2p quadrature accuracy and functional superconvergence in curvilinear coordinates when: (1) the Jacobian of the coordinate transformation is approximated by the same generalized summationbyparts operator that is used to approximate the flux terms and (2) the degree of the generalized summationbyparts operator is lower than the degree of the polynomial used to represent the geometry of interest. Legendrexe2x80x93Gaussxe2x80x93Lobatto and Legendrexe2x80x93Gauss elementtype operators are considered. When the aforementioned condition (2) is violated for the Legendrexe2x80x93Gauss operators, there is an evenxe2x80x93odd quadrature convergence pattern that is explained by the cancellation of the leading truncation error terms for the interpolationextrapolation operators that correspond to the odddegree Legendrexe2x80x93Gauss operators. The theory developed is confirmed through numerical examples with a steady onedimensional problem and the unsteady twodimensional linear convection equation.