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 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 »
- 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)
- OSTI Identifier:
- 1368024
- Report Number(s):
- LLNL-JRNL-676518
Journal ID: ISSN 1064-8275
- Grant/Contract Number:
- AC52-07NA27344
- Resource 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
- 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. https://doi.org/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. https://doi.org/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 = {Thu Sep 22 00:00:00 EDT 2016},
month = {Thu Sep 22 00:00:00 EDT 2016}
}
Web of Science
Works referenced in this record:
A Multiscale Mortar Mixed Finite Element Method
journal, January 2007
- Arbogast, Todd; Pencheva, Gergina; Wheeler, Mary F.
- Multiscale Modeling & Simulation, Vol. 6, Issue 1
Spectral AMGe ($\rho$AMGe)
journal, January 2003
- Chartier, T.; Falgout, R. D.; Henson, V. E.
- SIAM Journal on Scientific Computing, Vol. 25, Issue 1
A coupled local–global upscaling approach for simulating flow in highly heterogeneous formations
journal, October 2003
- Chen, Y.; Durlofsky, L. J.; Gerritsen, M.
- Advances in Water Resources, Vol. 26, Issue 10
A mixed multiscale finite element method for elliptic problems with oscillating coefficients
journal, June 2002
- Chen, Zhiming; Hou, Thomas Y.
- Mathematics of Computation, Vol. 72, Issue 242
Tenth SPE Comparative Solution Project: A Comparison of Upscaling Techniques
journal, August 2001
- Christie, M. A.; Blunt, M. J.
- SPE Reservoir Evaluation & Engineering, Vol. 4, Issue 04
Mixed Generalized Multiscale Finite Element Methods and Applications
journal, January 2015
- Chung, Eric T.; Efendiev, Yalchin; Lee, Chak Shing
- Multiscale Modeling & Simulation, Vol. 13, Issue 1
Numerical calculation of equivalent grid block permeability tensors for heterogeneous porous media
journal, May 1991
- Durlofsky, Louis J.
- Water Resources Research, Vol. 27, Issue 5
Robust domain decomposition preconditioners for abstract symmetric positive definite bilinear forms
journal, February 2012
- Efendiev, Yalchin; Galvis, Juan; Lazarov, Raytcho
- ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 46, Issue 5
Convergence of a Nonconforming Multiscale Finite Element Method
journal, January 2000
- Efendiev, Yalchin R.; Hou, Thomas Y.; Wu, Xiao-Hui
- SIAM Journal on Numerical Analysis, Vol. 37, Issue 3
Domain Decomposition Preconditioners for Multiscale Flows in High-Contrast Media
journal, January 2010
- Galvis, Juan; Efendiev, Yalchin
- Multiscale Modeling & Simulation, Vol. 8, Issue 4
Domain Decomposition Preconditioners for Multiscale Flows in High Contrast Media: Reduced Dimension Coarse Spaces
journal, January 2010
- Galvis, Juan; Efendiev, Yalchin
- Multiscale Modeling & Simulation, Vol. 8, Issue 5
A Multiscale Finite Element Method for Elliptic Problems in Composite Materials and Porous Media
journal, June 1997
- Hou, Thomas Y.; Wu, Xiao-Hui
- Journal of Computational Physics, Vol. 134, Issue 1
The variational multiscale method—a paradigm for computational mechanics
journal, November 1998
- Hughes, Thomas J. R.; Feijóo, Gonzalo R.; Mazzei, Luca
- Computer Methods in Applied Mechanics and Engineering, Vol. 166, Issue 1-2
The egg model - a geological ensemble for reservoir simulation
journal, November 2014
- Jansen, J. D.; Fonseca, R. M.; Kahrobaei, S.
- Geoscience Data Journal, Vol. 1, Issue 2
Multi-scale finite-volume method for elliptic problems in subsurface flow simulation
journal, May 2003
- Jenny, P.; Lee, S. H.; Tchelepi, H. A.
- Journal of Computational Physics, Vol. 187, Issue 1
AMGE Based on Element Agglomeration
journal, January 2001
- Jones, Jim E.; Vassilevski, Panayot S.
- SIAM Journal on Scientific Computing, Vol. 23, Issue 1
A Fast and High Quality Multilevel Scheme for Partitioning Irregular Graphs
journal, January 1998
- Karypis, George; Kumar, Vipin
- SIAM Journal on Scientific Computing, Vol. 20, Issue 1
A BDDC Algorithm with Enriched Coarse Spaces for Two-Dimensional Elliptic Problems with Oscillatory and High Contrast Coefficients
journal, January 2015
- Kim, Hyea Hyun; Chung, Eric T.
- Multiscale Modeling & Simulation, Vol. 13, Issue 2
On some versions of the element agglomeration AMGe method
journal, September 2008
- Lashuk, Ilya; Vassilevski, Panayot S.
- Numerical Linear Algebra with Applications, Vol. 15, Issue 7
Element agglomeration coarse Raviart-Thomas spaces with improved approximation properties: COARSE RAVIART-THOMAS SPACES WITH IMPROVED APPROXIMATION PROPERTIES
journal, January 2012
- Lashuk, I. V.; Vassilevski, P. S.
- Numerical Linear Algebra with Applications, Vol. 19, Issue 2
The Construction of the Coarse de Rham Complexes with Improved Approximation Properties
journal, January 2014
- Lashuk, Ilya V.; Vassilevski, Panayot S.
- Computational Methods in Applied Mathematics, Vol. 14, Issue 2
Preconditioning discretizations of systems of partial differential equations
journal, April 2010
- Mardal, Kent-Andre; Winther, Ragnar
- Numerical Linear Algebra with Applications, Vol. 18, Issue 1
Exact de Rham Sequences of Spaces Defined on Macro-Elements in Two and Three Spatial Dimensions
journal, January 2008
- Pasciak, Joseph E.; Vassilevski, Panayot S.
- SIAM Journal on Scientific Computing, Vol. 30, Issue 5
Abstract robust coarse spaces for systems of PDEs via generalized eigenproblems in the overlaps
journal, August 2013
- Spillane, N.; Dolean, V.; Hauret, P.
- Numerische Mathematik, Vol. 126, Issue 4
Solving generalized eigenvalue problems on the interfaces to build a robust two-level FETI method
journal, March 2013
- Spillane, Nicole; Dolean, Victorita; Hauret, Patrice
- Comptes Rendus Mathematique, Vol. 351, Issue 5-6
Coarse Spaces by Algebraic Multigrid: Multigrid Convergence and Upscaling Error Estimates
journal, April 2011
- Vassilevski, Panayot S.
- Advances in Adaptive Data Analysis, Vol. 03, Issue 01n02
Analysis of upscaling absolute permeability
journal, February 2002
- Wu, X. H.; Hou, Thomas; Efendiev, Y.
- Discrete and Continuous Dynamical Systems - Series B, Vol. 2, Issue 2
Works referencing / citing this record:
Scalable hierarchical PDE sampler for generating spatially correlated random fields using nonmatching meshes: Scalable hierarchical PDE sampler using nonmatching meshes
journal, January 2018
- Osborn, Sarah; Zulian, Patrick; Benson, Thomas
- Numerical Linear Algebra with Applications, Vol. 25, Issue 3
Multiscale two-stage solver for Biot’s poroelasticity equations in subsurface media
journal, November 2018
- Castelletto, Nicola; Klevtsov, Sergey; Hajibeygi, Hadi
- Computational Geosciences, Vol. 23, Issue 2
Scalable hierarchical PDE sampler for generating spatially correlated random fields using non-matching meshes
preprint, January 2017
- Osborn, Sarah; Zulian, Patrick; Benson, Thomas
- arXiv