  This paper is concerned with the design, analysis and implementation of
preconditioning concepts for spectral Discontinuous Galerkin discretizations of
elliptic boundary value problems. While presently known techniques realize a
growth of the condition numbers that is logarithmic in the polynomial degrees
when all degrees are equal and quadratic otherwise, our main objective is to
realize full robustness with respect to arbitrarily large locally varying
polynomial degrees degrees, i.e., under mild grading constraints condition
numbers stay uniformly bounded with respect to the mesh size and variable
degrees. The conceptual foundation of the envisaged preconditioners is the
auxiliary space method. The main conceptual ingredients that will be shown in
this framework to yield "optimal" preconditioners in the above sense are
Legendre-Gauss-Lobatto grids in connection with certain associated anisotropic
nested dyadic grids as well as specially adapted wavelet preconditioners for
the resulting low order auxiliary problems. Moreover, the preconditioners have
a modular form that facilitates somewhat simplified partial realizations. One
of the components can, for instance, be conveniently combined with domain
decomposition, at the expense though of a logarithmic growth of condition
numbers. Our analysis is complemented by quantitative experimental studies of
the main components.
