Multilevel preconditioning based on discrete symmetrization for convection-diffusion equations (Q1372103)

The authors consider a boundary value problem of the form \(-\nabla\cdot(\nabla u + \pmb{b} u) = f\) in \(\Omega \subset \mathbb{R}^d\) with homogeneous Dirichlet boundary condition on \(\partial\Omega\), where \(\Omega\) is a polygonal or polyhedral domain, \( d=2\) or \(3\), respectively, and \(\nabla\cdot\pmb{b} \in L^2( \Omega), f\in H^{-1}(\Omega)\). The subject of this paper is an additive multilevel preconditioning approach. The authors propose a modified multilevel preconditioner with improved convergence properties for convection-dominated problems which are discretized by the streamline diffusion method. For the constant coefficient case, an analysis of the convergence properties of the multilevel preconditioner is given in terms of its dependence on the convection size. Computational experiments support the theoretical results.