skip to main content
DOE PAGES title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: Efficient Operator-Coarsening Multigrid Schemes for Local Discontinuous Galerkin Methods

Abstract

An efficient hp-multigrid scheme is presented for local discontinuous Galerkin (LDG) discretizations of elliptic problems, formulated around the idea of separately coarsening the underlying discrete gradient and divergence operators. We show that traditional multigrid coarsening of the primal formulation leads to poor and suboptimal multigrid performance, whereas coarsening of the flux formulation leads to essentially optimal convergence and is equivalent to a purely geometric multigrid method. The resulting operator-coarsening schemes do not require the entire mesh hierarchy to be explicitly built, thereby obviating the need to compute quadrature rules, lifting operators, and other mesh-related quantities on coarse meshes. We show that good multigrid convergence rates are achieved in a variety of numerical tests on two-dimensional and three-dimensional uniform and adaptive Cartesian grids, as well as for curved domains using implicitly defined meshes and for multiphase elliptic interface problems with complex geometry. Extensions, e.g., to non-LDG discretizations and fully unstructured meshes, are briefly discussed.

Authors:
 [1];  [1];  [2]
  1. Harvard Univ., Cambridge, MA (United States)
  2. Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
Publication Date:
Research Org.:
Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
Sponsoring Org.:
USDOE Office of Science (SC)
OSTI Identifier:
1580384
Grant/Contract Number:  
AC02-05CH11231
Resource Type:
Accepted Manuscript
Journal Name:
SIAM Journal on Scientific Computing
Additional Journal Information:
Journal Volume: 41; Journal Issue: 6; Journal ID: ISSN 1064-8275
Publisher:
SIAM
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING

Citation Formats

Fortunato, Daniel, Rycroft, Chris H., and Saye, Robert. Efficient Operator-Coarsening Multigrid Schemes for Local Discontinuous Galerkin Methods. United States: N. p., 2019. Web. https://doi.org/10.1137/18m1206357.
Fortunato, Daniel, Rycroft, Chris H., & Saye, Robert. Efficient Operator-Coarsening Multigrid Schemes for Local Discontinuous Galerkin Methods. United States. https://doi.org/10.1137/18m1206357
Fortunato, Daniel, Rycroft, Chris H., and Saye, Robert. Tue . "Efficient Operator-Coarsening Multigrid Schemes for Local Discontinuous Galerkin Methods". United States. https://doi.org/10.1137/18m1206357. https://www.osti.gov/servlets/purl/1580384.
@article{osti_1580384,
title = {Efficient Operator-Coarsening Multigrid Schemes for Local Discontinuous Galerkin Methods},
author = {Fortunato, Daniel and Rycroft, Chris H. and Saye, Robert},
abstractNote = {An efficient hp-multigrid scheme is presented for local discontinuous Galerkin (LDG) discretizations of elliptic problems, formulated around the idea of separately coarsening the underlying discrete gradient and divergence operators. We show that traditional multigrid coarsening of the primal formulation leads to poor and suboptimal multigrid performance, whereas coarsening of the flux formulation leads to essentially optimal convergence and is equivalent to a purely geometric multigrid method. The resulting operator-coarsening schemes do not require the entire mesh hierarchy to be explicitly built, thereby obviating the need to compute quadrature rules, lifting operators, and other mesh-related quantities on coarse meshes. We show that good multigrid convergence rates are achieved in a variety of numerical tests on two-dimensional and three-dimensional uniform and adaptive Cartesian grids, as well as for curved domains using implicitly defined meshes and for multiphase elliptic interface problems with complex geometry. Extensions, e.g., to non-LDG discretizations and fully unstructured meshes, are briefly discussed.},
doi = {10.1137/18m1206357},
journal = {SIAM Journal on Scientific Computing},
number = 6,
volume = 41,
place = {United States},
year = {2019},
month = {12}
}

Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record

Save / Share: