How to gain speedups of 1000 on single processors with fast FEM solvers Benchmarking numerical and computational efficiency

In Computational Science and in particular in the numerical simulation of PDE problems, optimal serial performance is essential for a successful scale-out to the teraand petascale dimensions. In this paper, we propose a simple yet fundamental benchmark setting for a PDE problem that we believe any reasonably flexible Finite Element based software should be able to handle effortlessly. The Poisson problem used in these tests allows reliable performance estimates for more challenging simulations. Our performance evaluation focuses on numerical methodology and data layouts rather than implementational fine-tuning. To enable a fair and realistic comparison independent of the underlying numerical methodology, we define the metric total efficiency. Results are presented for two different solver classes, multigrid and Krylov-subspace methods, obtained in single-core computations with our solver packages Feat2 and Feast. We quantitatively emphasise the effect of different storage techniques and numbering (reordering) schemes, which constitute the crucial factor in view of the memory wall problem that ultimately determines performance of all Finite Element codes. We demonstrate a speed-up of more than a factor 1000 by migrating from a naive implementation of a standard Krylov solver to a sophisticated implementation of an advanced multigrid solver, without applying any adaptivity.