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Title: Upscaling of Mixed Finite Element Discretization Problems by the Spectral AMGe Method

Here, we propose two multilevel spectral techniques for constructing coarse discretization spaces for saddle-point problems corresponding to PDEs involving a divergence constraint, with a focus on mixed finite element discretizations of scalar self-adjoint 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 fine-grid 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 inf-sup 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) saddle-point 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 » Numerical experiments, in both two and three dimensions, are provided to illustrate the efficiency of the proposed upscaling technique.« less
Authors:
 [1] ;  [2] ;  [3] ;  [2] ;  [3]
  1. Univ. of Colorado, Boulder, CO (United States)
  2. Texas A & M Univ., College Station, TX (United States)
  3. Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
Publication Date:
Report Number(s):
LLNL-JRNL-676518
Journal ID: ISSN 1064-8275
Grant/Contract Number:
AC52-07NA27344
Type:
Accepted Manuscript
Journal Name:
SIAM Journal on Scientific Computing
Additional Journal Information:
Journal Volume: 38; Journal Issue: 5; Journal ID: ISSN 1064-8275
Publisher:
SIAM
Research Org:
Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
Sponsoring Org:
USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC-21)
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; element agglomeration; algebraic multigrid; spectral AMGe; upscaling; mixed finite elements
OSTI Identifier:
1368024

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., 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.. 2016. "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 saddle-point problems corresponding to PDEs involving a divergence constraint, with a focus on mixed finite element discretizations of scalar self-adjoint 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 fine-grid 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 inf-sup 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) saddle-point 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}
}