  The discontinuous Galerkin time-stepping method has many advantageous
properties for solving parabolic equations. However, it requires the solution
of a large nonsymmetric system at each time-step. This work develops a fully
robust and efficient preconditioning strategy for solving these systems.
Drawing on parabolic inf-sup theory, we first construct a left preconditioner
that transforms the linear system to a symmetric positive definite problem to
be solved by the preconditioned conjugate gradient algorithm. We then prove
that the transformed system can be further preconditioned by an ideal block
diagonal preconditioner, leading to a condition number bounded by 4 for any
time-step size, any approximation order and any positive-definite self-adjoint
spatial operators. Numerical experiments demonstrate the low condition numbers
and fast convergence of the algorithm for both ideal and approximate
preconditioners, and show the feasibility of the high-order solution of large
problems.
