Sparse Direct Factorizations through Unassembled Hyper-Matrices

We present a novel strategy for sparse direct factorizations that is geared towards the matrices that arise from hp-adaptive Finite Element Methods. In that context, a sequence of linear systems derived by successive local refinement of the problem domain needs to be solved. Thus, there is an opportunity for a factorization strategy that proceeds by updating (and possibly downdating) the factorization. Our scheme stores the matrix as unassembled element matrices, hierarchically ordered to mirror the refinement history of the domain. The factorization of such an ‘unassembled hyper-matrix’ proceeds in terms of element matrices, only assembling nodes when they need to be eliminated. The main benefits are efficiency from the fact that only updates to the factorization are made, high scalar efficiency since the factorization process uses dense matrices throughout, and a workflow that integrates naturally with the application. We present tests on 2D problems that bear out the large savings possible with hyper-matrix factorizations.