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Title: Approximate tensor-product preconditioners for very high order discontinuous Galerkin methods

Journal Article · · Journal of Computational Physics
ORCiD logo [1];  [2]
  1. Brown Univ., Providence, RI (United States). Div. of Applied Mathematics
  2. Univ. of California, Berkeley, CA (United States). Dept. of Mathematics
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In this study, we develop a new tensor-product based preconditioner for discontinuous Galerkin methods with polynomial degrees higher than those typically employed. This preconditioner uses an automatic, purely algebraic method to approximate the exact block Jacobi preconditioner by Kronecker products of several small, one-dimensional matrices. Traditional matrix-based preconditioners require O(p2d) storage and O(p3d) computational work, where p is the degree of basis polynomials used, and d is the spatial dimension. Our SVD-based tensor-product preconditioner requires O(pd+1) storage, O(pd+1) work in two spatial dimensions, and O(pd+2) work in three spatial dimensions. Combined with a matrix-free Newton–Krylov solver, these preconditioners allow for the solution of DG systems in linear time in p per degree of freedom in 2D, and reduce the computational complexity from O(p9) to O(p5) in 3D. Numerical results are shown in 2D and 3D for the advection, Euler, and Navier–Stokes equations, using polynomials of degree up to p = 30. For many test cases, the preconditioner results in similar iteration counts when compared with the exact block Jacobi preconditioner, and performance is significantly improved for high polynomial degrees p.

Research Organization:
Lawrence Berkeley National Laboratory (LBNL), Berkeley, CA (United States). National Energy Research Scientific Computing Center (NERSC)
Sponsoring Organization:
USDOE Office of Science (SC)
Grant/Contract Number:
AC02-05CH11231
OSTI ID:
1526501
Alternate ID(s):
OSTI ID: 1576607
Journal Information:
Journal of Computational Physics, Vol. 354, Issue C; ISSN 0021-9991
Publisher:
ElsevierCopyright Statement
Country of Publication:
United States
Language:
English
Citation Metrics:
Cited by: 27 works
Citation information provided by
Web of Science

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Cited By (12)

Analysis and Entropy Stability of the Line-Based Discontinuous Galerkin Method journal March 2019
A matrix-free high-order discontinuous Galerkin compressible Navier-Stokes solver: A performance comparison of compressible and incompressible formulations for turbulent incompressible flows: High-order matrix-free DG: Compressible vs incompressible journal October 2018
Analysis and entropy stability of the line-based discontinuous Galerkin method text January 2018
AutoHOOT: Automatic High-Order Optimization for Tensors
  • Ma, Linjian; Ye, Jiayu; Solomonik, Edgar
  • PACT '20: International Conference on Parallel Architectures and Compilation Techniques, Proceedings of the ACM International Conference on Parallel Architectures and Compilation Techniques https://doi.org/10.1145/3410463.3414647
conference September 2020
Automatic Code Generation for High-Performance Discontinuous Galerkin Methods on Modern Architectures preprint January 2018
Efficient low-order refined preconditioners for high-order matrix-free continuous and discontinuous Galerkin methods text January 2019
Fast Tensor Product Schwarz Smoothers for High-Order Discontinuous Galerkin Methods text January 2019
A high-order discontinuous Galerkin pressure robust splitting scheme for incompressible flows preprint January 2019
Sparse invariant domain preserving discontinuous Galerkin methods with subcell convex limiting text January 2020
An experimental comparison of a space-time multigrid method with PFASST for a reaction-diffusion problem preprint January 2020
Uniform subspace correction preconditioners for discontinuous Galerkin methods with $hp$-refinement text January 2020
Conservative and accurate solution transfer between high-order and low-order refined finite element spaces text January 2021

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Related Subjects

97 MATHEMATICS AND COMPUTING
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
Preconditioners
Discontinuous Galerkin method
Matrix-free