  We present and analyze a new space-time parallel multigrid method for
parabolic equations. The method is based on arbitrarily high order
discontinuous Galerkin discretizations in time, and a finite element
discretization in space. The key ingredient of the new algorithm is a block
Jacobi smoother. We present a detailed convergence analysis when the algorithm
is applied to the heat equation, and determine asymptotically optimal smoothing
parameters, a precise criterion for semi-coarsening in time or full coarsening,
and give an asymptotic two grid contraction factor estimate. We then explain
how to implement the new multigrid algorithm in parallel, and show with
numerical experiments its excellent strong and weak scalability properties.
