CLC in AMG solvers for saddle‐point problems

Abstract

An algebraic multigrid (AMG) solution method for saddle-point problems is described. The indefinite nature of the saddle-point matrix makes it unsuitable for the simple smoothing methods normally used in AMG. Moreover, even if presented in a stabilised form, as is done here, poorly conditioned matrices will be generated when constructing the coarse-grid approximation. This is because, with each successive coarsening step, the off-diagonal matrix blocks (of interfield coupling) reduce in size more slowly than the diagonal blocks (of intrafield coupling). Stabilised smoothing operators are therefore considered. The first is based on an incomplete decomposition of the complete system matrix into the product of lower-triangular (L) and upper-triangular (U) matrices, an ILU factorisation. The second is based on an exact block decomposition of an incomplete (simplified) system matrix into lower and upper block-triangular matrices, a BILU factorisation. However, the degree of stabilisation thus established in the smoothing operators does not guarantee an efficient smoothing at all grid levels. There can still be inefficiency on the least-stable coarser grids. The breakdown in efficiency begins at a grid level where the ratio of the inter-to intrafield coupling strengths exceeds a critical ratio. Provision is thus made for a further conditioning of coarse-grid operators, a coarse-level conditioning (CLC). This is another block-LU factorisation that is only applied at and beyond the critical grid level. It is not applied directly to the operator for any chosen level but to its fine-grid progenitor. Thus, while ILU and BILU use postcoarsening-step factorisations, CLC uses precoarsening-step factorisations. With CLC so deployed, ILU and BILU become efficient at all grid levels, resulting in an AMG convergence that is independent of the total number of grids.