Nodal auxiliary space preconditioning in H(curl) and H(div) spaces

In this paper, we develop and analyze a general approach to preconditioning linear systems of equations arising from conforming finite element discretizations of H(curl, Ω)- and H(div, Ω)-elliptic variational problems. The preconditioners exclusively rely on solvers for discrete Poisson problems. We prove mesh-independent effectivity of the preconditioners by using the abstract theory of auxiliary space preconditioning. The main tools are discrete analogues of so-called regular decomposition results in the function spaces H(curl, Ω) and H(div, Ω). Our preconditioner for H(curl, Ω) is similar to an algorithm proposed in [R. Beck, Algebraic Multigrid by Component Splitting for Edge Elements on Simplicial Triangulations, Tech. rep. SC 99-40, ZIB, Berlin, Germany, 1999].