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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 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, 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
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