Search Menu
DOE PAGES title logo U.S. Department of Energy
Office of Scientific and Technical Information
Search Scholarly Publications

Title: Monolithic Multigrid for a Reduced-Quadrature Discretization of Poroelasticity

Abstract

Advanced finite-element discretizations and preconditioners for models of poroelasticity have attracted significant attention in recent years. The equations of poroelasticity offer significant challenges in both areas, due to the potentially strong coupling between unknowns in the system, saddle-point structure, and the need to account for wide ranges of parameter values, including limiting behavior such as incompressible elasticity. This paper was motivated by an attempt to develop monolithic multigrid preconditioners for the discretization developed in [C. Rodrigo et al., Comput. Methods App. Mech. Engrg, 341 (2018), pp. 467--484]; we show here why this is a difficult task and, as a result, we modify the discretization in [Rodrigo et al.] through the use of a reduced-quadrature approximation, yielding a more “solver-friendly” discretization. Local Fourier analysis is used to optimize parameters in the resulting monolithic multigrid method, allowing a fair comparison between the performance and costs of methods based on Vanka and Braess--Sarazin relaxation. Further, numerical results are presented to validate the local Fourier analysis predictions and demonstrate efficiency of the algorithms. Finally, a comparison to existing block-factorization preconditioners is also given.

Authors:
ORCiD logo [1]; ORCiD logo [2]; ORCiD logo [1]; ORCiD logo [2];  [3]
+ Show Author Affiliations
  1. Tufts Univ., Medford, MA (United States)
  2. Memorial Univ. of Newfoundland, St. John's, NL (Canada)
  3. Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
Publication Date:
Research Org.:
Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
Sponsoring Org.:
USDOE National Nuclear Security Administration (NNSA); USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR); National Science Foundation (NSF)
OSTI Identifier:
1883184
Report Number(s):
SAND2022-5421J
Journal ID: ISSN 1064-8275; 705525
Grant/Contract Number:  
NA0003525; DMS-1620063
Resource Type:
Accepted Manuscript
Journal Name:
SIAM Journal on Scientific Computing
Additional Journal Information:
Journal Volume: 45; Journal Issue: 3; Journal ID: ISSN 1064-8275
Publisher:
Society for Industrial and Applied Mathematics (SIAM)
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; Biot poroelasticity; reduced quadrature; finite elements; monolithic multigrid; local Fourier analysis

Citation Formats

Adler, James H., He, Yunhui, Hu, Xiaozhe, MacLachlan, Scott, and Ohm, Peter. Monolithic Multigrid for a Reduced-Quadrature Discretization of Poroelasticity. United States: N. p., 2022. Web. doi:10.1137/21m1429072.
Copy to clipboard
Adler, James H., He, Yunhui, Hu, Xiaozhe, MacLachlan, Scott, & Ohm, Peter. Monolithic Multigrid for a Reduced-Quadrature Discretization of Poroelasticity. United States. https://doi.org/10.1137/21m1429072
Copy to clipboard
Adler, James H., He, Yunhui, Hu, Xiaozhe, MacLachlan, Scott, and Ohm, Peter. Fri . "Monolithic Multigrid for a Reduced-Quadrature Discretization of Poroelasticity". United States. https://doi.org/10.1137/21m1429072. https://www.osti.gov/servlets/purl/1883184.
Copy to clipboard
@article{osti_1883184,
title = {Monolithic Multigrid for a Reduced-Quadrature Discretization of Poroelasticity},
author = {Adler, James H. and He, Yunhui and Hu, Xiaozhe and MacLachlan, Scott and Ohm, Peter},
abstractNote = {Advanced finite-element discretizations and preconditioners for models of poroelasticity have attracted significant attention in recent years. The equations of poroelasticity offer significant challenges in both areas, due to the potentially strong coupling between unknowns in the system, saddle-point structure, and the need to account for wide ranges of parameter values, including limiting behavior such as incompressible elasticity. This paper was motivated by an attempt to develop monolithic multigrid preconditioners for the discretization developed in [C. Rodrigo et al., Comput. Methods App. Mech. Engrg, 341 (2018), pp. 467--484]; we show here why this is a difficult task and, as a result, we modify the discretization in [Rodrigo et al.] through the use of a reduced-quadrature approximation, yielding a more “solver-friendly” discretization. Local Fourier analysis is used to optimize parameters in the resulting monolithic multigrid method, allowing a fair comparison between the performance and costs of methods based on Vanka and Braess--Sarazin relaxation. Further, numerical results are presented to validate the local Fourier analysis predictions and demonstrate efficiency of the algorithms. Finally, a comparison to existing block-factorization preconditioners is also given.},
doi = {10.1137/21m1429072},
journal = {SIAM Journal on Scientific Computing},
number = 3,
volume = 45,
place = {United States},
year = {Fri Jun 24 00:00:00 EDT 2022},
month = {Fri Jun 24 00:00:00 EDT 2022}
}
Copy to clipboard
Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record
https://doi.org/10.1137/21m1429072
Copyright Statement
Other availability
Search WorldCat Search WorldCat to find libraries that may hold this journal
Save / Share:
Export Metadata
Save to My Library
You must Sign In or Create an Account in order to save documents to your library.

Works referenced in this record:

A partially parallel-in-time fixed-stress splitting method for Biot’s consolidation model
journal, March 2019

  • Borregales, Manuel; Kumar, Kundan; Radu, Florin Adrian
  • Computers & Mathematics with Applications, Vol. 77, Issue 6
  • DOI: 10.1016/j.camwa.2018.09.005

A nonconforming finite element method for the Biot’s consolidation model in poroelasticity
journal, January 2017

  • Hu, Xiaozhe; Rodrigo, Carmen; Gaspar, Francisco J.
  • Journal of Computational and Applied Mathematics, Vol. 310
  • DOI: 10.1016/j.cam.2016.06.003

On an Uzawa smoother in multigrid for poroelasticity equations: On an Uzawa smoother in multigrid for poroelasticity equations
journal, November 2016

  • Luo, P.; Rodrigo, C.; Gaspar, F. J.
  • Numerical Linear Algebra with Applications, Vol. 24, Issue 1
  • DOI: 10.1002/nla.2074

Efficient solvers for hybridized three-field mixed finite element coupled poromechanics
journal, June 2021

  • Frigo, Matteo; Castelletto, Nicola; Ferronato, Massimiliano
  • Computers & Mathematics with Applications, Vol. 91
  • DOI: 10.1016/j.camwa.2020.07.010

Local Fourier analysis for mixed finite-element methods for the Stokes equations
journal, September 2019

On a multigrid solver for the three-dimensional Biot poroelasticity system in multilayered domains
journal, March 2007

Multigrid in H (div) and H (curl)
journal, April 2000

  • Arnold, Douglas N.; Falk, Richard S.; Winther, Ragnar
  • Numerische Mathematik, Vol. 85, Issue 2
  • DOI: 10.1007/PL00005386

Stability and convergence of sequential methods for coupled flow and geomechanics: Fixed-stress and fixed-strain splits
journal, March 2011

  • Kim, J.; Tchelepi, H. A.; Juanes, R.
  • Computer Methods in Applied Mechanics and Engineering, Vol. 200, Issue 13-16
  • DOI: 10.1016/j.cma.2010.12.022

Convergence analysis of a new mixed finite element method for Biot's consolidation model
journal, February 2014

  • Yi, Son-Young
  • Numerical Methods for Partial Differential Equations, Vol. 30, Issue 4
  • DOI: 10.1002/num.21865

A local Fourier analysis of additive Vanka relaxation for the Stokes equations
journal, June 2020

  • Farrell, Patrick E.; He, Yunhui; MacLachlan, Scott P.
  • Numerical Linear Algebra with Applications, Vol. 28, Issue 3
  • DOI: 10.1002/nla.2306

On a local Fourier analysis for overlapping block smoothers on triangular grids
journal, July 2016

Asymptotic Behavior of Semidiscrete Finite-Element Approximations of Biot’s Consolidation Problem
journal, June 1996

  • Murad, Márcio A.; Thomée, Vidar; Loula, Abimael F. D.
  • SIAM Journal on Numerical Analysis, Vol. 33, Issue 3
  • DOI: 10.1137/0733052

Block preconditioners for mixed-dimensional discretization of flow in fractured porous media
journal, July 2020

Stability and monotonicity for some discretizations of the Biot’s consolidation model
journal, January 2016

  • Rodrigo, C.; Gaspar, F. J.; Hu, X.
  • Computer Methods in Applied Mechanics and Engineering, Vol. 298
  • DOI: 10.1016/j.cma.2015.09.019

Theory of Elasticity and Consolidation for a Porous Anisotropic Solid
journal, February 1955

Benchmarks for single-phase flow in fractured porous media
journal, January 2018

Theoretical Soil Mechanics
book, January 1943

New stabilized discretizations for poroelasticity and the Stokes’ equations
journal, November 2018

  • Rodrigo, C.; Hu, X.; Ohm, P.
  • Computer Methods in Applied Mechanics and Engineering, Vol. 341
  • DOI: 10.1016/j.cma.2018.07.003

Robust Error Analysis of Coupled Mixed Methods for Biot’s Consolidation Model
journal, April 2016

Consistent vs. reduced integration penalty methods for incompressible media using several old and new elements
journal, January 1982

  • Engelman, M. S.; Sani, R. L.; Gresho, P. M.
  • International Journal for Numerical Methods in Fluids, Vol. 2, Issue 1
  • DOI: 10.1002/fld.1650020103

Multigrid methods for a parameter dependent problem in primal variables
journal, November 1999

Scalable algorithms for three-field mixed finite element coupled poromechanics
journal, December 2016

  • Castelletto, Nicola; White, Joshua A.; Ferronato, Massimiliano
  • Journal of Computational Physics, Vol. 327
  • DOI: 10.1016/j.jcp.2016.09.063

Numerical performance of smoothers in coupled multigrid methods for the parallel solution of the incompressible Navier-Stokes equations
journal, January 2000

Improved accuracy in finite element analysis of Biot's consolidation problem
journal, March 1992

  • Murad, Márcio A.; Loula, Abimael F. D.
  • Computer Methods in Applied Mechanics and Engineering, Vol. 95, Issue 3
  • DOI: 10.1016/0045-7825(92)90193-N

Local Fourier analysis for multigrid with overlapping smoothers applied to systems of PDEs
journal, January 2011

  • MacLachlan, Scott P.; Oosterlee, C. W.
  • Numerical Linear Algebra with Applications, Vol. 18, Issue 4
  • DOI: 10.1002/nla.762

A comparative study of efficient iterative solvers for generalized Stokes equations
journal, January 2008

  • Larin, Maxim; Reusken, Arnold
  • Numerical Linear Algebra with Applications, Vol. 15, Issue 1
  • DOI: 10.1002/nla.561

Mixed finite element methods — Reduced and selective integration techniques: A unification of concepts
journal, July 1978

  • Malkus, David S.; Hughes, Thomas J. R.
  • Computer Methods in Applied Mechanics and Engineering, Vol. 15, Issue 1
  • DOI: 10.1016/0045-7825(78)90005-1

A stabilized hybrid mixed finite element method for poroelasticity
journal, July 2020

Weak Galerkin method for the Biot’s consolidation model
journal, March 2018

A high-order HDG method for the Biot’s consolidation model
journal, January 2019

A multigrid waveform relaxation method for solving the poroelasticity equations
journal, March 2018

  • Franco, S. R.; Rodrigo, C.; Gaspar, F. J.
  • Computational and Applied Mathematics, Vol. 37, Issue 4
  • DOI: 10.1007/s40314-018-0603-9

Tuning Multigrid Methods with Robust Optimization and Local Fourier Analysis
journal, January 2021

  • Brown, Jed; He, Yunhui; MacLachlan, Scott
  • SIAM Journal on Scientific Computing, Vol. 43, Issue 1
  • DOI: 10.1137/19M1308669

Accurate discretization of poroelasticity without Darcy stability
journal, March 2021

  • Mardal, Kent-Andre; Rognes, Marie E.; Thompson, Travis B.
  • BIT Numerical Mathematics, Vol. 61, Issue 3
  • DOI: 10.1007/s10543-021-00849-0

Unified approach to discretization of flow in fractured porous media
journal, November 2018

General Theory of Three‐Dimensional Consolidation
journal, February 1941

  • Biot, Maurice A.
  • Journal of Applied Physics, Vol. 12, Issue 2
  • DOI: 10.1063/1.1712886

Mixed Finite Element Methods and Applications
book, January 2013

A general preconditioning framework for coupled multiphysics problems with application to contact- and poro-mechanics
journal, December 2019

  • Ferronato, Massimiliano; Franceschini, Andrea; Janna, Carlo
  • Journal of Computational Physics, Vol. 398
  • DOI: 10.1016/j.jcp.2019.108887

Accuracy and convergence properties of the fixed-stress iterative solution of two-way coupled poromechanics: Accuracy and convergence properties of the fixed-stress iterative solution of two-way coupled poromechanics
journal, June 2015

  • Castelletto, N.; White, J. A.; Tchelepi, H. A.
  • International Journal for Numerical and Analytical Methods in Geomechanics, Vol. 39, Issue 14
  • DOI: 10.1002/nag.2400

Robust Preconditioners for a New Stabilized Discretization of the Poroelastic Equations
journal, January 2020

  • Adler, J. H.; Gaspar, F. J.; Hu, X.
  • SIAM Journal on Scientific Computing, Vol. 42, Issue 3
  • DOI: 10.1137/19M1261250

On the fixed-stress split scheme as smoother in multigrid methods for coupling flow and geomechanics
journal, November 2017

  • Gaspar, Francisco J.; Rodrigo, Carmen
  • Computer Methods in Applied Mechanics and Engineering, Vol. 326
  • DOI: 10.1016/j.cma.2017.08.025

Preconditioning a mass‐conserving discontinuous Galerkin discretization of the Stokes equations
journal, March 2016

  • Adler, James H.; Benson, Thomas R.; MacLachlan, Scott P.
  • Numerical Linear Algebra with Applications, Vol. 24, Issue 3
  • DOI: 10.1002/nla.2047

A Study of Two Modes of Locking in Poroelasticity
journal, January 2017

  • Yi, Son-Young
  • SIAM Journal on Numerical Analysis, Vol. 55, Issue 4
  • DOI: 10.1137/16M1056109

Two‐level Fourier analysis of multigrid for higher‐order finite‐element discretizations of the Laplacian
journal, February 2020

  • He, Yunhui; MacLachlan, Scott
  • Numerical Linear Algebra with Applications, Vol. 27, Issue 3
  • DOI: 10.1002/nla.2285

Block-implicit multigrid calculation of two-dimensional recirculating flows
journal, November 1986

A Simple and Efficient Segregated Smoother for the Discrete Stokes Equations
journal, January 2014

  • Gaspar, Francisco J.; Notay, Yvan; Oosterlee, Cornelis W.
  • SIAM Journal on Scientific Computing, Vol. 36, Issue 3
  • DOI: 10.1137/130920630

An efficient smoother for the Stokes problem
journal, February 1997

On stability and convergence of finite element approximations of Biot's consolidation problem
journal, February 1994

  • Murad, Márcio A.; Loula, Abimael F. D.
  • International Journal for Numerical Methods in Engineering, Vol. 37, Issue 4
  • DOI: 10.1002/nme.1620370407

Parameter-robust stability of classical three-field formulation of Biot's consolidation model
journal, January 2018

  • Hong, Qingguo; Kraus, Johannes
  • ETNA - Electronic Transactions on Numerical Analysis, Vol. 48
  • DOI: 10.1553/etna_vol48s202

A coupling of nonconforming and mixed finite element methods for Biot's consolidation model
journal, February 2013

  • Yi, Son-Young
  • Numerical Methods for Partial Differential Equations, Vol. 29, Issue 5
  • DOI: 10.1002/num.21775

Block-partitioned solvers for coupled poromechanics: A unified framework
journal, May 2016

  • White, Joshua A.; Castelletto, Nicola; Tchelepi, Hamdi A.
  • Computer Methods in Applied Mechanics and Engineering, Vol. 303
  • DOI: 10.1016/j.cma.2016.01.008
    Sort by:
    [ × clear filter / sort ]

    Similar Records in DOE PAGES and OSTI.GOV collections: