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New Stabilized Discretizations for Poroelasticity Equations [chapter]

Francisco J. Gaspar, Carmen Rodrigo, Xiaozhe Hu, Peter Ohm, James Adler, Ludmil Zikatanov
2019 Lecture Notes in Computer Science  
The difference is in the mechanics: one of the discretizations uses Crouzeix-Raviart nonconforming linear elements; the other is based on piecewise linear elements stabilized by using face bubbles, which  ...  They employ the lowestorder mixed finite elements for the flow (Raviart-Thomas-Nédélec elements for the Darcy velocity and piecewise constants for the pressure) and are stable with respect to the physical  ...  Regarding the numerical simulation of the poroelasticity equations, there have been numerous contributions using finitedifference schemes [6, 7] and finite-volume methods (see [8, 9] for recent developments  ... 
doi:10.1007/978-3-030-10692-8_1 fatcat:rm52u2ihs5fvznw2pswvqocpp4

New stabilized discretizations for poroelasticity and the Stokes' equations

C. Rodrigo, X. Hu, P. Ohm, J.H. Adler, F.J. Gaspar, L.T. Zikatanov
2018 Computer Methods in Applied Mechanics and Engineering  
We prove optimal stability and error estimates for this discretization.  ...  Numerical tests confirm the theory for both poroelastic and Stokes' test problems.  ...  A consequence of our analysis is that a new stable scheme for the Stokes' equations is derived.  ... 
doi:10.1016/j.cma.2018.07.003 fatcat:wd2a6e6k5zcqrdak7ylsjcobay

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 last equation has been scaled by τ for symmetry. To simplify the notation, we carry out the following stability analysis for a constant time-step size.  ... 
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 last equation has been scaled by \tau for symmetry. To simplify the notation, we carry out the following stability analysis for a constant time-step size.  ... 
doi:10.1137/19m1261250 fatcat:c3pm4cu3frdddnthw2catnxzhy

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.  ...  We emphasize the saddle point type aspect in these two systems and discuss their discretization on staggered and collocated grids.  ...  Collocated poroelasticity discretization We deal with the incompressible poroelasticity equations, in which a stabilization term ε∂ p/∂t [15] , with ε = h 2 /4(λ + 2μ) is added: −μ˜ u − (λ + μ) grad div  ... 
doi:10.1016/j.apnum.2007.11.014 fatcat:kvhdqqqmtbf5rdeyiemh2ezwry

A stabilized difference scheme for deformable porous media and its numerical resolution by multigrid methods

F. J. Gaspar, F. J. Lisbona, C. W. Oosterlee
2007 Computing and Visualization in Science  
This paper deals with the 2D system of incompressible poroelasticity equations in which an artificial stabilization term has been added to the discretization on collocated grids.  ...  Secondly, various smoothers are examined in order to find an optimal multigrid method for the discrete system of equations.  ...  Stabilization is done by adding an extra term in one of the continuous equations. It has been proved that a stable and accurate solution is obtained with this new set of poroelastic equations.  ... 
doi:10.1007/s00791-007-0061-1 fatcat:vowz6smq5fdqxevpzxynk333w4

An efficient multigrid solver for a reformulated version of the poroelasticity system

F.J. Gaspar, F.J. Lisbona, C.W. Oosterlee, P.N. Vabishchevich
2007 Computer Methods in Applied Mechanics and Engineering  
In this paper, we present a robust and efficient multigrid solver for a reformulated version of the system of poroelasticity equations.  ...  We show that the reformulation boils down to a stabilization term in the iterative scheme, and that the solution of the original problem is identical to the solution of the reformulated problem.  ...  So far, we can prove this stability issue only for the 1D poroelasticity model situation. 3 One-Dimensional Poroelasticity 1D Transformation In 1D, the governing equations simplify and read (P) : − (  ... 
doi:10.1016/j.cma.2006.03.020 fatcat:ebnph5lvabe2rng7qbcwjk6q2i

Diffraction of plane P waves by a canyon of arbitrary shape in poroelastic half-space (I): Formulation

Jianwen Liang, Zhongxian Liu
2009 Earthquake Science  
This paper presents an indirect boundary integration equation method for diffraction of plane P waves by a two-dimensional canyon of arbitrary shape in poroelastic half-space.  ...  The Green's functions of compressional and shear wave sources in poroelastic half-space are derived based on Biot's theory.  ...  Acknowledgements The authors gratefully acknowledge support from the Program for New Century Excellent Talents in University (NCET-05-0248) and the Key Program for Applied Basic Research of Tianjin Municipality  ... 
doi:10.1007/s11589-009-0215-y fatcat:7ngarx3verde7mzu2ky7highc4

Splitting schemes for poroelasticity and thermoelasticity problems [article]

A.E. Kolesov, P.N. Vabishchevich, M.V. Vasilyeva
2013 arXiv   pre-print
Stability estimates of weighted difference schemes for the coupled system of equations are presented. Approximation in space is based on the finite element method.  ...  In this work, we consider the coupled systems of linear unsteady partial differential equations, which arise in the modeling of poroelasticity processes.  ...  Stability estimates of weighted schemes for the coupled system of equations are obtained for the differential and discrete problem using Samarskii's theory of stability for operator-difference schemes.  ... 
arXiv:1311.3766v1 fatcat:p3fl4k37hnesfjwgwszw3u7o5m

Multigrid method for nonlinear poroelasticity equations

P. Luo, C. Rodrigo, F. J. Gaspar, C. W. Oosterlee
2015 Computing and Visualization in Science  
For the unsteady problem, an additional artificial term is utilized to stabilize the solutions when the equations are discretized on collocated grids.  ...  In this study, a nonlinear multigrid method is applied for solving the system of incompressible poroelasticity equations considering nonlinear hydraulic conductivity.  ...  Acknowledgments Peiyao Luo is grateful for the financial support by the China Scholarship Council (CSC).  ... 
doi:10.1007/s00791-016-0260-8 fatcat:ogqhpx2bi5bell5trf2quao2ce

Hybrid Fixed-Point Fixed-Stress Splitting Method for Linear Poroelasticity

Paul M. Delgado, V. M. Krushnarao Kotteda, Vinod Kumar
2019 Geosciences  
Three different cases are considered to demonstrate the stability and consistency of the method for constant and variable parameters.  ...  We propose an efficient operator splitting method for Biot's model of linear poroelasticity based on fixed-point iteration and constrained stress.  ...  Acknowledgments: We thank two anonymous reviewers for comments that improved the manuscript.  ... 
doi:10.3390/geosciences9010029 fatcat:h6hs4zaokfehpddpw3kof55av4

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.  ...  For poroelasticity we evaluated the Vanka smoothers, in [15] for staggered grid discretizations, and in [17] for stabilized vertex-centered discretizations.  ...  For example, Olshanskii and Reusken showed in [31] that the grad-div term has a stabilization effect on the discrete Stokes equations. This process was called "graddiv-stabilization".  ... 
doi:10.1002/nla.587 fatcat:dgua4ewq5fdyfj3am4awv3ad3e

Finite-difference modeling of wave propagation and diffusion in poroelastic media

Fabian Wenzlau, Tobias M. Müller
2009 Geophysics  
A series of numerical experiments that are compared to exact analytical solutions allow us to assess the stability conditions and dispersion relations of the explicit poroelastic finite-difference method  ...  To confirm and further develop rockphysics theories for reservoir rocks, accurate numerical modeling tools are required.  ...  ACKNOWLEDGMENTS The poroelastic finite-difference program presented here has been developed based on a parallelized 2D viscoelastic algorithm whose author, Thomas Bohlen, is acknowledged for providing  ... 
doi:10.1190/1.3122928 fatcat:s2t7m4nuyndo5k6b3i4fqeneke

An unsplit convolutional perfectly matched layer improved at grazing incidence for seismic wave propagation in poroelastic media

Roland Martin, Dimitri Komatitsch, Abdelâziz Ezziani
2008 Geophysics  
We develop a PML improved at grazing incidence for the poroelastic wave equation based on an unsplit convolutional formulation of the equation as a first-order system in velocity and stress.  ...  However, after numerical discretization, at grazing incidence, large spurious oscillations are sent back from the PML into the main domain.  ...  Gedney for fruitful discussions and for providing them with his CPML software package for Maxwell's equations and Julien Diaz for providing them with the analytical solution for the first validation test  ... 
doi:10.1190/1.2939484 fatcat:w7rlvmtagvbaxdv3ridzrv3lce

Defmod - Parallel multiphysics finite element code for modeling crustal deformation during the earthquake/rifting cycle [article]

S. Tabrez Ali
2015 arXiv   pre-print
Results are written in ASCII VTK format for easy visualization.  ...  Problems can be solved using (stabilized) linear triangular, quadrilateral, tetrahedral or hexahedral elements on shared or distributed memory machines with hundreds or even thousands of processor cores  ...  Kyriakopoulos for his help in validating Defmod, and (iv) Kurt Feigl and Herb Wang for support and constructive suggestions.  ... 
arXiv:1402.0429v3 fatcat:nzwdxx6dhjamlow4aa472et7oe
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