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

Abstract

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.

Authors:
ORCiD logo [1];  [2]
+ Show Author Affiliations
  1. Brown Univ., Providence, RI (United States). Div. of Applied Mathematics
  2. Univ. of California, Berkeley, CA (United States). Dept. of Mathematics
Publication Date:
Research Org.:
Lawrence Berkeley National Laboratory (LBNL), Berkeley, CA (United States). National Energy Research Scientific Computing Center (NERSC)
Sponsoring Org.:
USDOE Office of Science (SC)
OSTI Identifier:
1526501
Alternate Identifier(s):
OSTI ID: 1576607
Grant/Contract Number:  
AC02-05CH11231
Resource Type:
Accepted Manuscript
Journal Name:
Journal of Computational Physics
Additional Journal Information:
Journal Volume: 354; Journal Issue: C; Journal ID: ISSN 0021-9991
Publisher:
Elsevier
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Preconditioners; Discontinuous Galerkin method; Matrix-free

Citation Formats

Pazner, Will, and Persson, Per -Olof. Approximate tensor-product preconditioners for very high order discontinuous Galerkin methods. United States: N. p., 2017. Web. doi:10.1016/j.jcp.2017.10.030.
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Pazner, Will, & Persson, Per -Olof. Approximate tensor-product preconditioners for very high order discontinuous Galerkin methods. United States. https://doi.org/10.1016/j.jcp.2017.10.030
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Pazner, Will, and Persson, Per -Olof. Fri . "Approximate tensor-product preconditioners for very high order discontinuous Galerkin methods". United States. https://doi.org/10.1016/j.jcp.2017.10.030. https://www.osti.gov/servlets/purl/1526501.
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@article{osti_1526501,
title = {Approximate tensor-product preconditioners for very high order discontinuous Galerkin methods},
author = {Pazner, Will and Persson, Per -Olof},
abstractNote = {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.},
doi = {10.1016/j.jcp.2017.10.030},
journal = {Journal of Computational Physics},
number = C,
volume = 354,
place = {United States},
year = {2017},
month = {11}
}
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