A sub-grid scale finite element agglomeration multigrid method with application to the Boltzmann transport equation

This article describes a new element agglomeration multigrid method for solving partial differential equations discretised through a sub-grid scale finite element formulation. The sub-grid scale discretisation resolves solution variables through their separate coarse and fine scales, and these are mapped between the multigrid levels using a dual set of transfer operators. The sub-grid scale multigrid method forms coarse linear systems, possessing the same sub-grid scale structure as the original discretisation, that can be resolved without them being stored in memory. This is necessary for the application of this article in resolving the Boltzmann transport equation as the linear systems become extremely large. The novelty of this article is therefore a matrix-free multigrid scheme that is integrated within its own sub-grid scale discretisation using dual transfer operators and applied to the Boltzmann transport equation. The numerical examples presented are designed to show the method's preconditioning capabilities for a Krylov space-based solver. The problems range in difficulty, geometry and discretisation type, and comparisons made with established methods show this new approach to perform consistently well. Smoothing operators are also analysed and this includes using the generalized minimal residual method. Here, it is shown that an adaptation to the preconditioned Krylov space is necessary for it to work efficiently. Copyright (c) 2012 John Wiley & Sons, Ltd.