TY - JOUR
T1 - Efficient matrix-free high-order finite element evaluation for simplicial elements
AU - Moxey, David
AU - Amici, Roman
AU - Kirby, Mike
N1 - Funding Information:
\ast Submitted to the journal's Software and High-Performance Computing section February 25, 2019; accepted for publication (in revised form) March 16, 2020; published electronically May 26, 2020. https://doi.org/10.1137/19M1246523 \bfF \bfu \bfn \bfd \bfi \bfn \bfg : The work of the first author was supported by the PRISM project through EPSRC under grant EP/R029423/1. The work of the second and third authors was supported by the AFRL through grant FA8650-17-C-5269. \dagger College of Engineering, Mathematics and Physical Sciences, University of Exeter, Exeter, Devon, EX17 1EJ, UK (d.moxey@exeter.ac.uk). \ddagger Scientific Computing and Imaging Institute, University of Utah, Salt Lake City, UT 84112 (amicir@gmail.com, kirby@cs.utah.edu).
Publisher Copyright:
© 2020 Societ y for Industrial and Applied Mathematics.
Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.
PY - 2020
Y1 - 2020
N2 - With the gap between processor clock speeds and memory bandwidth speeds continuing to increase, the use of arithmetically intense schemes, such as high-order finite element methods, continues to be of considerable interest. In particular, the use of matrix-free formulations of finite element operators for tensor-product elements of quadrilaterals in two dimensions and hexahedra in three dimensions, in combination with single-instruction multiple-data instruction sets, is a well-studied topic at present for the efficient implicit solution of elliptic equations. However, a considerable limiting factor for this approach is the use of meshes comprising of only quadrilaterals or hexahedra, the creation of which is still an open problem within the mesh generation community. In this article, we study the efficiency of high-order finite element operators for the Helmholtz equation with a focus on extending this approach to unstructured meshes of triangles, tetrahedra, and prismatic elements using the spectral/hp element method and corresponding tensor-product bases for these element types. We show that although performance is naturally degraded when going from hexahedra to these simplicial elements, efficient implementations can still be obtained that are capable of attaining 50% through 70% floating point operations of the peak of processors with both AVX2 and AVX512 instruction sets.
AB - With the gap between processor clock speeds and memory bandwidth speeds continuing to increase, the use of arithmetically intense schemes, such as high-order finite element methods, continues to be of considerable interest. In particular, the use of matrix-free formulations of finite element operators for tensor-product elements of quadrilaterals in two dimensions and hexahedra in three dimensions, in combination with single-instruction multiple-data instruction sets, is a well-studied topic at present for the efficient implicit solution of elliptic equations. However, a considerable limiting factor for this approach is the use of meshes comprising of only quadrilaterals or hexahedra, the creation of which is still an open problem within the mesh generation community. In this article, we study the efficiency of high-order finite element operators for the Helmholtz equation with a focus on extending this approach to unstructured meshes of triangles, tetrahedra, and prismatic elements using the spectral/hp element method and corresponding tensor-product bases for these element types. We show that although performance is naturally degraded when going from hexahedra to these simplicial elements, efficient implementations can still be obtained that are capable of attaining 50% through 70% floating point operations of the peak of processors with both AVX2 and AVX512 instruction sets.
KW - High-order finite elements
KW - High-performance computing
KW - SIMD vectorization
KW - Spectral/hp element method
UR - http://www.scopus.com/inward/record.url?scp=85094139023&partnerID=8YFLogxK
U2 - 10.1137/19M1246523
DO - 10.1137/19M1246523
M3 - Article
AN - SCOPUS:85094139023
SN - 1064-8275
VL - 42
SP - C97-C123
JO - SIAM JOURNAL ON SCIENTIFIC COMPUTING
JF - SIAM JOURNAL ON SCIENTIFIC COMPUTING
IS - 3
ER -