We present a novel approach to fast on-the-fly low order finite element assembly for scalar elliptic partial differential equations of Darcy type with variable coefficients optimized for matrix-free implementations. Our approach introduces a new operator that is obtained by appropriately scaling the reference stiffness matrix from the constant coefficient case. Assuming sufficient regularity, an a priori analysis shows that solutions obtained by this approach are unique and have asymptotically optimal order convergence in the H1- and the L2-norm on hierarchical hybrid grids. For the pre-asymptotic regime, we present a local modification that guarantees uniform ellipticity of the operator. Cost considerations show that our novel approach requires roughly one third of the floating-point operations compared to a classical finite element assembly scheme employing nodal integration. Our theoretical considerations are illustrated by numerical tests that confirm the expectations with respect to accuracy and run-time. A large scale application with more than a hundred billion (1.6⋅1011) degrees of freedom executed on 14,310 compute cores demonstrates the efficiency of the new scaling approach.
@article{arxiv.1709.06793,
title = {A stencil scaling approach for accelerating matrix-free finite element implementations},
author = {Simon Bauer and Daniel Drzisga and Marcus Mohr and Ulrich Ruede and Christian Waluga and Barbara Wohlmuth},
journal= {arXiv preprint arXiv:1709.06793},
year = {2018}
}