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Robust preconditioners for a new stabilized discretization of the poroelastic equations [article]

James H. Adler, Francisco J. Gaspar, Xiaozhe Hu, Peter Ohm, Carmen Rodrigo, Ludmil T. Zikatanov
2020 arXiv   pre-print
In this paper, we present block preconditioners for a stabilized discretization of the poroelastic equations developed in [45].  ...  We construct both norm-equivalent (diagonal) and field-of-value-equivalent (triangular) preconditioners for both the stabilized discretization and a perturbation of the stabilized discretization that leads  ...  The stability and well-posedness of the discrete problem provides a foundation for designing robust preconditioners.  ... 
arXiv:1905.10353v2 fatcat:wa6jejrl6be6hog6jajc56lntm

Robust Preconditioners for a New Stabilized Discretization of the Poroelastic Equations

J. H. Adler, F. J. Gaspar, X. Hu, P. Ohm, C. Rodrigo, L. T. Zikatanov
2020 SIAM Journal on Scientific Computing  
In this paper, we present block preconditioners for a stabilized discretization of the poroelastic equations developed in [C. Rodrigo, X. Hu, P. Ohm, J. Adler, F. Gaspar, and L. Zikatanov, Comput.  ...  We construct both norm-equivalent (diagonal) and field-of-value-equivalent (triangular) preconditioners for both the stabilized discretization and a perturbation of the stabilized discretization, which  ...  The stability and well-posedness of the discrete problem provide a foundation for designing robust preconditioners.  ... 
doi:10.1137/19m1261250 fatcat:c3pm4cu3frdddnthw2catnxzhy

Robust block preconditioners for poroelasticity [article]

Shuangshuang Chen and Qingguo Hong and Jinchao Xu and Kai Yang
2020 arXiv   pre-print
Numerical tests demonstrate the robustness of these preconditioners.  ...  In this paper we study the linear systems arising from discretized poroelasticity problems.  ...  Robust preconditioner for a new three-field formulation introducing a total pressure as the third unknown is analyzed in [28] .  ... 
arXiv:2001.08254v1 fatcat:2ylottua4re3zbwfmq7ccngoyu

Conservative discretizations and parameter-robust preconditioners for Biot and multiple-network flux-based poroelastic models [article]

Qinggou Hong and Johannes Kraus and Maria Lymbery and Fadi Philo
2018 arXiv   pre-print
model, we prove the uniform stability, and design stable disretizations and parameter-robust preconditioners for flux-based formulations of multiple-network poroelastic systems.  ...  The parameters in the governing system of partial differential equations of multicompartmental poroelastic models typically vary over several orders of magnitude making its stable discretization and efficient  ...  Biot's model of poroelasticity combines these equations and the parameter-robust stability of its classical three-field formulation has been established only recently in [34] .  ... 
arXiv:1806.00353v2 fatcat:zz3o7rgozrhlznqv5sdyyvmfg4

Parameter robust preconditioning by congruence for multiple-network poroelasticity [article]

Eleonora Piersanti, Jeonghun J. Lee, Travis Thompson, Kent-Andre Mardal, Marie E. Rognes
2020 arXiv   pre-print
In particular, we introduce a new approach to formulating finite element methods and associated preconditioners for the MPET equations.  ...  In this paper, we present a new strategy for efficiently and robustly solving the MPET equations numerically.  ...  In this section we discuss a new formulation for the MPT equations, which in turn easily offers a parameter-robust preconditioner.  ... 
arXiv:2003.09641v2 fatcat:b3j5w7znzvgtxialvjvnahw7mi

New multigrid smoothers for the Oseen problem

Steven Hamilton, Michele Benzi, Eldad Haber
2010 Numerical Linear Algebra with Applications  
We investigate the performance of smoothers based on the Hermitian/skew-Hermitian (HSS) and augmented Lagrangian (AL) splittings applied to MAC discretizations of the Oseen problem.  ...  Local Fourier analysis and numerical experiments on a 2-D lid-driven cavity problem indicate that the proposed smoothers result in hindependent convergence and are fairly robust with respect to the Reynolds  ...  It is of interest to note that the (1,1) block of the augmented system (3.1) resembles that of the poroelasticity equations described in [15] (though the poroelasticity equations lack a convective term  ... 
doi:10.1002/nla.707 fatcat:vufppcep55gqtgry3xetuhmsb4

Multigrid relaxation methods for systems of saddle point type

C.W. Oosterlee, F.J. Gaspar
2008 Applied Numerical Mathematics  
In this paper, we give an overview of multigrid methods for two systems of equations, namely the Stokes equations and the incompressible poroelasticity equations.  ...  The basic problem is that of smoothing a system of equations that has a zero (or almost zero) block in the matrix for one of the unknowns.  ...  For poroelasticity on staggered grids, coupled relaxation was not fully robust, especially not regarding the choice of the time step, whereas distributive relaxation on the staggered grid was robust for  ... 
doi:10.1016/j.apnum.2007.11.014 fatcat:kvhdqqqmtbf5rdeyiemh2ezwry

A Stabilized Hybrid Mixed Finite Element Method for Poroelasticity [article]

Chunyan Niu and Hongxing Rui and Xiaozhe Hu
2020 arXiv   pre-print
We introduce a perturbation of the bilinear form of the displacement which allows for the elimination of the bubble functions.  ...  Based on the well-posedness of the eliminated system, we design block preconditioners that are parameter-robust.  ...  In such case, the discrete problem approaches a P1-P0 discretization of the Stokes' equation.  ... 
arXiv:1906.00885v2 fatcat:neduz5xohfaz5nv5gnokh7vo4m

Parameter Robust Preconditioning by Congruence for Multiple-Network Poroelasticity

E. Piersanti, J. J. Lee, T. Thompson, K.-A. Mardal, M. E. Rognes
2021 SIAM Journal on Scientific Computing  
In this paper, we present a new strategy for efficiently and robustly solving the MPET equations numerically.  ...  The mechanical behavior of a poroelastic medium permeated by multiple interacting fluid networks can be described by a system of time-dependent partial differential equations known as the multiple-network  ...  In this example we demonstrate the robustness of the block diagonal preconditioner (4.23) for a mixed finite element discretization of the transformed total pressure MPET equations (4.16).  ... 
doi:10.1137/20m1326751 fatcat:7j65va67tzgsld7f55ybacyx2m

Monolithic multigrid for a reduced-quadrature discretization of poroelasticity [article]

James H. Adler, Yunhui He, Xiaozhe Hu, Scott MacLachlan, Peter Ohm
2022 arXiv   pre-print
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  ...  After discretization, large linear systems of equations must be solved to compute the finite-element approximation to the solution of the poroelasticity equations.  ... 
arXiv:2107.04060v3 fatcat:qhq5h3oigvhp5hpt3hrenfaz5q

Distributive smoothers in multigrid for problems with dominating grad-div operators

F. J. Gaspar, J. L. Gracia, F. J. Lisbona, C. W. Oosterlee
2008 Numerical Linear Algebra with Applications  
In this paper we present efficient multigrid methods for systems of partial differential equations that are governed by a dominating grad-div operator.  ...  In particular, we show that distributive smoothing methods give multigrid convergence factors that are independent of problem parameters and of the mesh sizes in space and time.  ...  In [1] , Arnold et al. presented a multigrid preconditioner for a discrete system of equations which arises when discretizing the variational formulation of a grad-div equation with the lowest order Raviart-Thomas  ... 
doi:10.1002/nla.587 fatcat:dgua4ewq5fdyfj3am4awv3ad3e

A parallel block preconditioner for large-scale poroelasticity with highly heterogeneous material parameters

Joachim Berdal Haga, Harald Osnes, Hans Petter Langtangen
2012 Computational Geosciences  
We present a parallel preconditioner for Biot's equations of coupled elasticity and fluid flow in porous media.  ...  The approximation relies on a highly scalable calculation of the global Schur complement of the coefficient matrix, combined with generally available state-of-the-art multilevel preconditioners for the  ...  Paper IV: A parallel block preconditioner for large scale poroelasticity with highly heterogeneous material parameters In Paper II, we developed and tested robust block preconditioners for Biot's equations  ... 
doi:10.1007/s10596-012-9284-4 fatcat:3l2y2qnnjfe7pazsikzmslrozy

Parameter-robust discretization and preconditioning of Biot's consolidation model [article]

Jeonghun J. Lee, Kent-Andre Mardal, Ragnar Winther
2016 arXiv   pre-print
The purpose of this paper is to study finite element discretizations of this model and construct block diagonal preconditioners for the discrete Biot systems.  ...  We derive preconditioners that are robust with respect to both the variations of the parameters and the mesh refinement.  ...  Parameter-robust stability of the continuous problems For the rest of this paper we will discuss a system of the form (2.7), where the parameter µ ′ = 1.  ... 
arXiv:1507.03199v4 fatcat:3utvttip2vg3tc7pzfalotpyxa

Parameter-Robust Discretization and Preconditioning of Biot's Consolidation Model

Jeonghun J. Lee, Kent-Andre Mardal, Ragnar Winther
2017 SIAM Journal on Scientific Computing  
The purpose of this paper is to study finite element discretizations of this model and construct block diagonal preconditioners for the discrete Biot systems.  ...  We derive preconditioners that are robust with respect to both the variations of the parameters and the mesh refinement.  ...  Parameter-robust stability of the continuous problems For the rest of this paper we will discuss a system of the form (2.7), where the parameter µ = 1.  ... 
doi:10.1137/15m1029473 fatcat:ujohl5lldnhp3ahk55uue37vle

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

Francisco J. Gaspar, Carmen Rodrigo
2017 Computer Methods in Applied Mechanics and Engineering  
In this work, we analyze the behaviour of an inexact version of this algorithm as smoother within a geometric multigrid method, in order to obtain an efficient monolithic solver for the Biot's problem.  ...  This solver combines the advantages of being a fully coupled method, with the benefit of decoupling the flow and the mechanics part in the smoothing algorithm.  ...  Acknowledgements The work of Francisco J. Gaspar is supported by the European Union's  ... 
doi:10.1016/j.cma.2017.08.025 fatcat:7o6hxgvi3rdj5dhg2rps3e6nre
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