  We present a static-condensation method for time-implicit discretizations of
the Discontinuous Galerkin Spectral Element Method on Gauss-Lobatto points
(GL-DGSEM). We show that, when solving the compressible Navier-Stokes
equations, it is possible to reorganize the linear system that results from the
implicit time-integration of the GL-DGSEM as a Schur complement problem, which
can be efficiently solved using static condensation. The use of static
condensation reduces the linear system size and improves the condition number
of the system matrix, which translates into shorter computational times when
using direct and iterative solvers.
  The statically condensed GL-DGSEM presented here can be applied to linear and
nonlinear advection-diffusion partial differential equations in conservation
form. To test it we solve the compressible Navier-Stokes equations with direct
and Krylov subspace solvers, and we show for a selected problem that using the
statically condensed GL-DGSEM leads to speed-ups of up to $200$ when compared
to the time-explicit GL-DGSEM, and speed-ups of up to three when compared with
the time-implicit GL-DGSEM that solves the global system.
  The GL-DGSEM has gained increasing popularity in recent years because it
satisfies the summation-by-parts property, which enables the construction of
provably entropy stable schemes, and because it is computationally very
efficient. In this paper, we show that the GL-DGSEM has an additional
advantage: It can be statically condensed.
