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

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.

@article{id2411,
  author = {Bauer, S. and Drzisga, D. and Mohr, M. and R\"ude, U. and Waluga, C. and Wohlmuth, B.},
  doi = {10.1137/17M1148384},
  journal = {SIAM Journal on Scientific Computing},
  language = {en},
  number = {6},
  pages = {C748{\textendash}C778},
  title = {A stencil scaling approach for accelerating matrix-free finite element implementations},
  url = {https://www.geophysik.uni-muenchen.de/{\textasciitilde}mohr/Publikationen/bauer\_2018\_sisc.pdf},
  volume = {40},
  year = {2018},
}

%O Journal Article
%A Bauer, S.
%A Drzisga, D.
%A Mohr, M.
%A Rüde, U.
%A Waluga, C.
%A Wohlmuth, B.
%R 10.1137/17M1148384
%J SIAM Journal on Scientific Computing
%G en
%N 6
%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
%V 40
%D 2018