A stencil scaling approach for accelerating matrix-free finite element implementations

Bauer, S., D. Drzisga, M. Mohr, U. Rüde, C. Waluga, and B. Wohlmuth (2018), A stencil scaling approach for accelerating matrix-free finite element implementations, SIAM Journal on Scientific Computing, 40(6), C748–C778, doi:10.1137/17M1148384.

Abstract

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 $H^1$- and the $L^2$-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 ($1.6\cdot10^11$) degrees of freedom executed on 14\,310 compute cores demonstrates the efficiency of the new scaling approach.

Further information

• bauer_2018_sisc.pdf

BibTeX

@article{id2411,
  author = {S. Bauer and D. Drzisga and M. Mohr and U. R{\"u}de and C. Waluga and B. Wohlmuth},
  journal = {SIAM Journal on Scientific Computing},
  number = {6},
  pages = {C748{--}C778},
  title = {{A stencil scaling approach for accelerating matrix-free finite element implementations}},
  volume = {40},
  year = {2018},
  language = {en},
  url = {https://www.geophysik.uni-muenchen.de/~mohr/Publikationen/bauer{\_}2018{\_}sisc.pdf},
  doi = {10.1137/17M1148384},
}

EndNote

%0 Journal Article
%A Bauer, S.
%A Drzisga, D.
%A Mohr, M.
%A Rüde, U.
%A Waluga, C.
%A Wohlmuth, B.
%D 2018
%N 6
%V 40
%J SIAM Journal on Scientific Computing
%P C748–C778
%T A stencil scaling approach for accelerating matrix-free finite element implementations
%U https://www.geophysik.uni-muenchen.de/~mohr/Publikationen/bauer_2018_sisc.pdf