On an adaptive multigrid solver for convection-dominated problems

The objective of this paper is to develop and describe an adaptive multigrid scheme for the numerical solution of convection-diffusion-reaction equations with dominating convection or reaction term. Specifically, we propose a systematic construction of problem and scale-dependent restriction, prolongation and smoothing operators in combination with an adaptive mesh refinement strategy. In particular, adaptivity is not only to reduce computational complexity but also to stabilize standard Galerkin discretizations so that adding artificial viscosity can be avoided. The approach is suited for any choice of underlying multiresolution sequences, in particular for standard finite element discretizations on nested but possibly nonuniform meshes. Moreover, all concepts are in principle independent of the spatial dimension and avoid any special flow-dependent enumeration of unknowns. The performance of the scheme is illustrated by various examples in one and two spatial dimensions. They are chosen so as to exhibit possibly many types of adverse or singular behavior in connection with constant and varying convection or the interplay between convective and reactive terms. In the light of these tests those issues are identified that require further research.