Upscaling of Mixed Finite Element Discretization Problems by the Spectral AMGe Method
Abstract
Here, we propose two multilevel spectral techniques for constructing coarse discretization spaces for saddlepoint problems corresponding to PDEs involving a divergence constraint, with a focus on mixed finite element discretizations of scalar selfadjoint second order elliptic equations on general unstructured grids. We use element agglomeration algebraic multigrid (AMGe), which employs coarse elements that can have nonstandard shape since they are agglomerates of finegrid elements. The coarse basis associated with each agglomerated coarse element is constructed by solving local eigenvalue problems and local mixed finite element problems. This construction leads to stable upscaled coarse spaces and guarantees the infsup compatibility of the upscaled discretization. Also, the approximation properties of these upscaled spaces improve by adding more local eigenfunctions to the coarse spaces. The higher accuracy comes at the cost of additional computational effort, as the sparsity of the resulting upscaled coarse discretization (referred to as operator complexity) deteriorates when we introduce additional functions in the coarse space. We also provide an efficient solver for the coarse (upscaled) saddlepoint system by employing hybridization, which leads to a symmetric positive definite (s.p.d.) reduced system for the Lagrange multipliers, and to solve the latter s.p.d. system, we use our previously developed spectral AMGe solver.more »
 Authors:

 Univ. of Colorado, Boulder, CO (United States)
 Texas A & M Univ., College Station, TX (United States)
 Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
 Publication Date:
 Research Org.:
 Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
 Sponsoring Org.:
 USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC21)
 OSTI Identifier:
 1368024
 Report Number(s):
 LLNLJRNL676518
Journal ID: ISSN 10648275
 Grant/Contract Number:
 AC5207NA27344
 Resource Type:
 Accepted Manuscript
 Journal Name:
 SIAM Journal on Scientific Computing
 Additional Journal Information:
 Journal Volume: 38; Journal Issue: 5; Journal ID: ISSN 10648275
 Publisher:
 SIAM
 Country of Publication:
 United States
 Language:
 English
 Subject:
 97 MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; element agglomeration; algebraic multigrid; spectral AMGe; upscaling; mixed finite elements
Citation Formats
Kalchev, Delyan Z., Lee, C. S., Villa, U., Efendiev, Y., and Vassilevski, P. S. Upscaling of Mixed Finite Element Discretization Problems by the Spectral AMGe Method. United States: N. p., 2016.
Web. doi:10.1137/15M1036683.
Kalchev, Delyan Z., Lee, C. S., Villa, U., Efendiev, Y., & Vassilevski, P. S. Upscaling of Mixed Finite Element Discretization Problems by the Spectral AMGe Method. United States. doi:10.1137/15M1036683.
Kalchev, Delyan Z., Lee, C. S., Villa, U., Efendiev, Y., and Vassilevski, P. S. Thu .
"Upscaling of Mixed Finite Element Discretization Problems by the Spectral AMGe Method". United States. doi:10.1137/15M1036683. https://www.osti.gov/servlets/purl/1368024.
@article{osti_1368024,
title = {Upscaling of Mixed Finite Element Discretization Problems by the Spectral AMGe Method},
author = {Kalchev, Delyan Z. and Lee, C. S. and Villa, U. and Efendiev, Y. and Vassilevski, P. S.},
abstractNote = {Here, we propose two multilevel spectral techniques for constructing coarse discretization spaces for saddlepoint problems corresponding to PDEs involving a divergence constraint, with a focus on mixed finite element discretizations of scalar selfadjoint second order elliptic equations on general unstructured grids. We use element agglomeration algebraic multigrid (AMGe), which employs coarse elements that can have nonstandard shape since they are agglomerates of finegrid elements. The coarse basis associated with each agglomerated coarse element is constructed by solving local eigenvalue problems and local mixed finite element problems. This construction leads to stable upscaled coarse spaces and guarantees the infsup compatibility of the upscaled discretization. Also, the approximation properties of these upscaled spaces improve by adding more local eigenfunctions to the coarse spaces. The higher accuracy comes at the cost of additional computational effort, as the sparsity of the resulting upscaled coarse discretization (referred to as operator complexity) deteriorates when we introduce additional functions in the coarse space. We also provide an efficient solver for the coarse (upscaled) saddlepoint system by employing hybridization, which leads to a symmetric positive definite (s.p.d.) reduced system for the Lagrange multipliers, and to solve the latter s.p.d. system, we use our previously developed spectral AMGe solver. Numerical experiments, in both two and three dimensions, are provided to illustrate the efficiency of the proposed upscaling technique.},
doi = {10.1137/15M1036683},
journal = {SIAM Journal on Scientific Computing},
number = 5,
volume = 38,
place = {United States},
year = {2016},
month = {9}
}
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