  This work concerns the analysis of the discontinuous Galerkin spectral
element method (DGSEM) with implicit time stepping for the numerical
approximation of nonlinear scalar conservation laws in multiple space
dimensions. We consider either the DGSEM with a backward Euler time stepping,
or a space-time DGSEM discretization to remove the restriction on the time
step. We design graph viscosities in space, and in time for the space-time
DGSEM, to make the schemes maximum principle preserving and entropy stable for
every admissible convex entropy. We also establish well-posedness of the
discrete problems by showing existence and uniqueness of the solutions to the
nonlinear implicit algebraic relations that need to be solved at each time
step. Numerical experiments in one space dimension are presented to illustrate
the properties of these schemes.
