Algebraic Multigrid for Linear Systems Obtained by Explicit Element Reduction

• Authors: Thomas A. Brunner;Tzanio V. Kolev
• Venue: SIAM Journal on Scientific Computing
• Year: 2011

We consider sparse linear systems, where a set of “interior” degrees of freedom have been eliminated in order to reduce the problem size. This elimination is assumed to be local, so the “interior” principal submatrix is block-diagonal, and the resulting Schur complement is still sparse. For it to be beneficial, the elimination process should lead to reduced memory requirements, but equally importantly, it should also result in an algebraic problem that can be solved efficiently. In this paper we propose a general element reduction approach and show how the elimination process can exploit a particular “subzonal” discretization to maintain the sparsity of the Schur complement. We also investigate algebraic multigrid (AMG) solution algorithms applied to the reduced problem, and we discuss the influence of the local elimination on solver-related properties of the matrix, such as near-nullspace preservation and the availability of stable subspace decompositions. We focus on BoomerAMG, a parallel variant of classical Ruge-Stüben AMG, applied to scalar diffusion problems [V. Henson and U. Yang, Appl. Numer. Math., 41 (2002), pp. 155-177], and the auxiliary-space Maxwell solver (AMS) for electromagnetic diffusion applications [T. Kolev and P. Vassilevski, J. Comput. Math., 27 (2009), pp. 604-623]. In the electromagnetic case, we establish algebraically a reduced version of the Hiptmair-Xu decomposition from [R. Hiptmair and J. Xu, SIAM J. Numer. Anal., 45 (2007), pp. 2483-2509] and consider a modification of the reduction process that targets the singular problems arising in simulations with pure void (zero conductivity) regions. For scalar diffusion problems, our particular stencil analysis shows that the reduction has a positive effect on meshes with stretched elements. We present a number of two-dimensional, three-dimensional, and axisymmetric numerical experiments, which demonstrate that the combination of an appropriately chosen local elimination with the use of the BoomerAMG and AMS solvers can lead to significant improvements in the overall solution time.