Robust norm equivalencies for diffusion problems

Additive multilevel methods offer an efficient way for the fast solution of large sparse linear systems which arise from a finite element discretization of an elliptic boundary value problem. These solution methods are based on multilevel norm equivalencies for the associated bilinear form using a suitable subspace decomposition. To obtain a robust iterative scheme, it is crucial that the constants in the norm equivalence do not depend or depend only weakly on the ellipticity constants of the problem. In this paper we present such a robust norm equivalence for the model problem $- \nabla \omega \nabla u=f$ with a scalar diffusion coefficient $\omega$ in $\Omega \subset \mathbb {R}^2$. Our estimates involve only very weak information about $\omega$, and the results are applicable for a large class of diffusion coefficients. Namely, we require $\omega$ to be in the Muckenhoupt class $A_{1}(\Omega )$, a function class well-studied in harmonic analysis. The presented multilevel norm equivalencies are a main step towards the realization of an optimal and robust multilevel preconditioner for scalar diffusion problems.