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This paper is concerned with the design, analysis and implementation ofpreconditioning concepts for spectral Discontinuous Galerkin discretizations ofelliptic boundary value problems. While presently known techniques realize agrowth of the condition numbers that is logarithmic in the polynomial degreeswhen all degrees are equal and quadratic otherwise, our main objective is torealize full robustness with respect to arbitrarily large locally varyingpolynomial degrees degrees, i.e., under mild grading constraints conditionnumbers stay uniformly bounded with respect to the mesh size and variabledegrees. The conceptual foundation of the envisaged preconditioners is theauxiliary space method. The main conceptual ingredients that will be shown inthis framework to yield "optimal" preconditioners in the above sense areLegendre-Gauss-Lobatto grids in connection with certain associated anisotropicnested dyadic grids as well as specially adapted wavelet preconditioners forthe resulting low order auxiliary problems. Moreover, the preconditioners havea modular form that facilitates somewhat simplified partial realizations. Oneof the components can, for instance, be conveniently combined with domaindecomposition, at the expense though of a logarithmic growth of conditionnumbers. Our analysis is complemented by quantitative experimental studies ofthe main components.

Multilevel Preconditioning of Discontinuous-Galerkin Spectral Element Methods, Part I: Geometrically Conforming Meshes