Algebraic multigrid and discrete calculus representations of coupled-fields

This paper discusses the application of scalar algebraic multigrid methods to the solution of discrete approximations to coupled scalar-vector fields. All discretizations considered are based on the direct enforcement of the conservation laws on chosen control volumes. They are distinguished by different choices for the field variables. Both local (point based) and non-local (domain based) variables are considered; these include variables associated with volumes (cells), areas (faces), lines (edges) and points (nodes) of the computational mesh. Two algebraic multigrid (AMG) methods are applied; both are based on the scalar unknown approach of Ruge and Stueben, but distinguished by exploiting different Galerkin coarse grid approximations (CGAs). The first is the classical method (C-AMG) in which the CGAs are based on system reduction by subset selection. The second is the method of Vanek and others, in which the CGA is based on system reduction by aggregation (SA-AMG). Clearly, for consistent coarse-grid approximations, the equations must be capable of being sensibly combined in the case of SA-AMG, and that chosen subsets be sensibly representative in the case of C-AMG. By addressing different discrete formulations of the Stokes problem, the investigation shows that some formulations, which may be convergent using single-grid solution methods, may not be convergent when using scalar AMG solvers. This is because they do not allow the coarsening criteria mentioned earlier to be satisfied without additional explicit geometrical/topological information, that is, the equation system alone does not contain sufficient information to permit the construction of consistent CGAs. The findings remind us that, for consistent CGAs, the coarsening process should always respect both the equation type (scalar, vector, etc) and the topology of those space elements that are an integral part of the field description (oriented points, lines, areas and volumes). A more general approach to AMG seems to be required for discrete calculus applications, possibly based on the methods of algebraic topology. Copyright (C) 2011 John Wiley & Sons, Ltd.