Nested dyadic grids associated with Legendre–Gauss–Lobatto grids

Abstract

Legendre---Gauss---Lobatto (LGL) grids play a pivotal role in nodal spectral methods for the numerical solution of partial differential equations. They not only provide efficient high-order quadrature rules, but give also rise to norm equivalences that could eventually lead to efficient preconditioning techniques in high-order methods. Unfortunately, a serious obstruction to fully exploiting the potential of such concepts is the fact that LGL grids of different degree are not nested. This affects, on the one hand, the choice and analysis of suitable auxiliary spaces, when applying the auxiliary space method as a principal preconditioning paradigm, and, on the other hand, the efficient solution of the auxiliary problems. As a central remedy, we consider certain nested hierarchies of dyadic grids of locally comparable mesh size, that are in a certain sense properly associated with the LGL grids. Their actual suitability requires a subtle analysis of such grids which, in turn, relies on a number of refined properties of LGL grids. The central objective of this paper is to derive the main properties of the associated dyadic grids needed for preconditioning the systems arising from $$hp$$hp- or even spectral (conforming or Discontinuous Galerkin type) discretizations for second order elliptic problems in a way that is fully robust with respect to varying polynomial degrees. To establish these properties requires revisiting some refined properties of LGL grids and their relatives.