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Hybridized weak Galerkin finite element methods for Brinkman equations
Decoupling PDE computation with intrinsic or inertial Robin interface condition
1. | Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hongkong, China |
2. | Department of Mathematics, Morgan State University, 1700 E Cold Spring Ln, MD 21251, USA |
We study decoupled numerical methods for multi-domain, multi-physics applications. By investigating various stages of numerical approximation and decoupling and tracking how the information is transmitted across the interface for a typical multi-modeling model problem, we derive an approximate intrinsic or inertial type Robin condition for its semi-discrete model. This new interface condition is justified both mathematically and physically in contrast to the classical Robin interface condition conventionally introduced for decoupling multi-modeling problems. Based on the intrinsic or inertial Robin condition, an equivalent semi-discrete model is introduced, which provides a general framework for devising effective decoupled numerical methods. Numerical experiments also confirm the effectiveness of this new decoupling approach.
References:
[1] |
L. Badea, M. Discacciati and A. Quarteroni,
Numerical analysis of the Navier–Stokes/Darcy coupling, Numer. Math., 115 (2010), 195-227.
doi: 10.1007/s00211-009-0279-6. |
[2] |
S. Badia, F. Nobile and C. Vergara,
Fluid–structure partitioned procedures based on Robin transmission conditions, J. Comput. Phys., 227 (2008), 7027-7051.
doi: 10.1016/j.jcp.2008.04.006. |
[3] |
A. T. Barker and X.-C. Cai,
Scalable parallel methods for monolithic coupling in fluid–structure interaction with application to blood flow modeling, J. Comput. Phys., 229 (2010), 642-659.
doi: 10.1016/j.jcp.2009.10.001. |
[4] |
M. Bukac and B. Muha,
Stability and convergence analysis of the extensions of the kinematically coupled scheme for the fluid-structure interaction, SIAM J. Numer. Anal., 54 (2016), 3032-3061.
doi: 10.1137/16M1055396. |
[5] |
M. Cai, M. Mu and J. Xu,
Numerical solution to a mixed Navier–Stokes/Darcy model by the two-grid approach, SIAM J. Numer. Anal., 47 (2009), 3325-3338.
doi: 10.1137/080721868. |
[6] |
M. Cai, M. Mu and J. Xu,
Preconditioning techniques for a mixed Stokes/Darcy model in porous media applications, J. Comput. Appl. Math., 233 (2009), 346-355.
doi: 10.1016/j.cam.2009.07.029. |
[7] |
S. Canic, B. Muha and M. Bukac,
Stability of the kinematically coupled $\beta$-scheme for fluid-structure interaction problems in hemodynamics, Int. J. Numer. Anal. Model., 12 (2015), 54-80.
|
[8] |
Y. Cao, M. Gunzburger, F. Hua and X. Wang,
Coupled Stokes-Darcy model with Beavers-Joseph interface boundary condition, Commun. Math. Sci., 8 (2010), 1-25.
doi: 10.4310/CMS.2010.v8.n1.a2. |
[9] |
P. Causin, J.-F. Gerbeau and F. Nobile,
Added-mass effect in the design of partitioned algorithms for fluid–structure problems, Comput. Methods Appl. Mech. Engrg., 194 (2005), 4506-4527.
doi: 10.1016/j.cma.2004.12.005. |
[10] |
Z. Chen and J. Zou,
Finite element methods and their convergence for elliptic and parabolic interface problems, Numer. Math., 79 (1998), 175-202.
doi: 10.1007/s002110050336. |
[11] |
P. Crosetto, P. Reymond, S. Deparis, D. Kontaxakis, N. Stergiopulos and A. Quarteroni,
Fluid–structure interaction simulation of aortic blood flow, Comput. & Fluids, 43 (2011), 46-57.
doi: 10.1016/j.compfluid.2010.11.032. |
[12] |
M. Discacciati and A. Quarteroni,
Navier-Stokes/Darcy coupling: Modeling, analysis, and numerical approximation, Rev. Mat. Complut., 22 (2009), 315-426.
doi: 10.5209/rev_REMA.2009.v22.n2.16263. |
[13] |
M. Discacciati, A. Quarteroni and A. Valli,
Robin–Robin domain decomposition methods for the Stokes–Darcy coupling, SIAM J. Numer. Anal., 45 (2007), 1246-1268.
doi: 10.1137/06065091X. |
[14] |
M. A. Fernández, J. Mullaert and M. Vidrascu,
Explicit Robin–Neumann schemes for the coupling of incompressible fluids with thin-walled structures, Comput. Methods Appl. Mech. Engrg., 267 (2013), 566-593.
doi: 10.1016/j.cma.2013.09.020. |
[15] |
M. A. Fernández, J. Mullaert and M. Vidrascu,
Generalized Robin–Neumann explicit coupling schemes for incompressible fluid-structure interaction: Stability analysis and numerics, Internat. J. Numer. Methods Engrg., 101 (2015), 199-229.
doi: 10.1002/nme.4785. |
[16] |
C. Förster, W. A. Wall and E. Ramm,
Artificial added mass instabilities in sequential staggered coupling of nonlinear structures and incompressible viscous flows, Comput. Methods Appl. Mech. Engrg., 196 (2007), 1278-1293.
doi: 10.1016/j.cma.2006.09.002. |
[17] |
R. Lan, P. Sun and M. Mu,
Mixed finite element analysis for an elliptic/mixed elliptic interface problem with jump coefficients, Procedia Comput. Sci., 108 (2017), 1913-1922.
doi: 10.1016/j.procs.2017.05.001. |
[18] |
W. Leng, C.-S. Zhang, P. Sun, B. Gao and J. Xu,
Numerical simulation of an immersed rotating structure in fluid for hemodynamic applications, J. Comput. Sci., 30 (2019), 79-89.
doi: 10.1016/j.jocs.2018.11.010. |
[19] |
M. Mu and J. Xu,
A two-grid method of a mixed Stokes–Darcy model for coupling fluid flow with porous media flow, SIAM J. Numer. Anal., 45 (2007), 1801-1813.
doi: 10.1137/050637820. |
[20] |
M. Mu and X. Zhu,
Decoupled schemes for a non-stationary mixed Stokes-Darcy model, Math. Comp., 79 (2010), 707-731.
doi: 10.1090/S0025-5718-09-02302-3. |
[21] |
A. Quarteroni and A. Valli, Domain Decomposition Methods for Partial Differential Equations, Numerical Mathematics and Scientific Computation, The Clarendon Press, Oxford University Press, New York, 1999. |
[22] |
M. Razzaq, H. Damanik, J. Hron, A. Ouazzi and S. Turek,
FEM multigrid techniques for fluid-structure interaction with application to hemodynamics, Appl. Numer. Math., 62 (2012), 1156-1170.
doi: 10.1016/j.apnum.2010.12.010. |
[23] |
B. Rivière,
Analysis of a discontinuous finite element method for the coupled Stokes and Darcy problems, J. Sci. Comput., 22/23 (2005), 479-500.
doi: 10.1007/s10915-004-4147-3. |
[24] |
B. Rivière and I. Yotov,
Locally conservative coupling of Stokes and Darcy flows, SIAM J. Numer. Anal., 42 (2005), 1959-1977.
doi: 10.1137/S0036142903427640. |
[25] |
R. K. Sinha and B. Deka,
Optimal error estimates for linear parabolic problems with discontinuous coefficients, SIAM J. Numer. Anal., 43 (2005), 733-749.
doi: 10.1137/040605357. |
[26] |
P. Sun and C. Wang,
Distributed Lagrange multiplier/fictitious domain finite element method for Stokes/parabolic interface problems with jump coefficients, Appl. Numer. Math., 152 (2020), 199-220.
doi: 10.1016/j.apnum.2019.12.009. |
[27] |
V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Springer Series in Computational Mathematics, 25, Springer-Verlag, Berlin, 2006.
doi: 10.1007/3-540-33122-0. |
[28] |
S. Turek and J. Hron, Proposal for numerical benchmarking of fluid-structure interaction between an elastic object and laminar incompressible flow, in Fluid Structure Interaction, Lect. Notes Comput. Sci. Eng., 53, Springer, Berlin, 2006,371–385.
doi: 10.1007/3-540-34596-5_15. |
[29] |
S. Turek, J. Hron, M. Razzaq, H. Wobker and M. Schäfer, Numerical benchmarking of fluid-structure interaction: A comparison of different discretization and solution approaches, In Fluid Structure Interaction II, Lect. Notes Comput. Sci. Eng., 73, Springer, Heidelberg, 2010,413–424.
doi: 10.1007/978-3-642-14206-2_15. |
[30] |
T. Wick,
Solving monolithic fluid-structure interaction problems in arbitrary Lagrangian Eulerian coordinates with the deal. II Library, Arch. Numer. Software, 1 (2013), 1-19.
doi: 10.11588/ans.2013.1.10305. |
show all references
References:
[1] |
L. Badea, M. Discacciati and A. Quarteroni,
Numerical analysis of the Navier–Stokes/Darcy coupling, Numer. Math., 115 (2010), 195-227.
doi: 10.1007/s00211-009-0279-6. |
[2] |
S. Badia, F. Nobile and C. Vergara,
Fluid–structure partitioned procedures based on Robin transmission conditions, J. Comput. Phys., 227 (2008), 7027-7051.
doi: 10.1016/j.jcp.2008.04.006. |
[3] |
A. T. Barker and X.-C. Cai,
Scalable parallel methods for monolithic coupling in fluid–structure interaction with application to blood flow modeling, J. Comput. Phys., 229 (2010), 642-659.
doi: 10.1016/j.jcp.2009.10.001. |
[4] |
M. Bukac and B. Muha,
Stability and convergence analysis of the extensions of the kinematically coupled scheme for the fluid-structure interaction, SIAM J. Numer. Anal., 54 (2016), 3032-3061.
doi: 10.1137/16M1055396. |
[5] |
M. Cai, M. Mu and J. Xu,
Numerical solution to a mixed Navier–Stokes/Darcy model by the two-grid approach, SIAM J. Numer. Anal., 47 (2009), 3325-3338.
doi: 10.1137/080721868. |
[6] |
M. Cai, M. Mu and J. Xu,
Preconditioning techniques for a mixed Stokes/Darcy model in porous media applications, J. Comput. Appl. Math., 233 (2009), 346-355.
doi: 10.1016/j.cam.2009.07.029. |
[7] |
S. Canic, B. Muha and M. Bukac,
Stability of the kinematically coupled $\beta$-scheme for fluid-structure interaction problems in hemodynamics, Int. J. Numer. Anal. Model., 12 (2015), 54-80.
|
[8] |
Y. Cao, M. Gunzburger, F. Hua and X. Wang,
Coupled Stokes-Darcy model with Beavers-Joseph interface boundary condition, Commun. Math. Sci., 8 (2010), 1-25.
doi: 10.4310/CMS.2010.v8.n1.a2. |
[9] |
P. Causin, J.-F. Gerbeau and F. Nobile,
Added-mass effect in the design of partitioned algorithms for fluid–structure problems, Comput. Methods Appl. Mech. Engrg., 194 (2005), 4506-4527.
doi: 10.1016/j.cma.2004.12.005. |
[10] |
Z. Chen and J. Zou,
Finite element methods and their convergence for elliptic and parabolic interface problems, Numer. Math., 79 (1998), 175-202.
doi: 10.1007/s002110050336. |
[11] |
P. Crosetto, P. Reymond, S. Deparis, D. Kontaxakis, N. Stergiopulos and A. Quarteroni,
Fluid–structure interaction simulation of aortic blood flow, Comput. & Fluids, 43 (2011), 46-57.
doi: 10.1016/j.compfluid.2010.11.032. |
[12] |
M. Discacciati and A. Quarteroni,
Navier-Stokes/Darcy coupling: Modeling, analysis, and numerical approximation, Rev. Mat. Complut., 22 (2009), 315-426.
doi: 10.5209/rev_REMA.2009.v22.n2.16263. |
[13] |
M. Discacciati, A. Quarteroni and A. Valli,
Robin–Robin domain decomposition methods for the Stokes–Darcy coupling, SIAM J. Numer. Anal., 45 (2007), 1246-1268.
doi: 10.1137/06065091X. |
[14] |
M. A. Fernández, J. Mullaert and M. Vidrascu,
Explicit Robin–Neumann schemes for the coupling of incompressible fluids with thin-walled structures, Comput. Methods Appl. Mech. Engrg., 267 (2013), 566-593.
doi: 10.1016/j.cma.2013.09.020. |
[15] |
M. A. Fernández, J. Mullaert and M. Vidrascu,
Generalized Robin–Neumann explicit coupling schemes for incompressible fluid-structure interaction: Stability analysis and numerics, Internat. J. Numer. Methods Engrg., 101 (2015), 199-229.
doi: 10.1002/nme.4785. |
[16] |
C. Förster, W. A. Wall and E. Ramm,
Artificial added mass instabilities in sequential staggered coupling of nonlinear structures and incompressible viscous flows, Comput. Methods Appl. Mech. Engrg., 196 (2007), 1278-1293.
doi: 10.1016/j.cma.2006.09.002. |
[17] |
R. Lan, P. Sun and M. Mu,
Mixed finite element analysis for an elliptic/mixed elliptic interface problem with jump coefficients, Procedia Comput. Sci., 108 (2017), 1913-1922.
doi: 10.1016/j.procs.2017.05.001. |
[18] |
W. Leng, C.-S. Zhang, P. Sun, B. Gao and J. Xu,
Numerical simulation of an immersed rotating structure in fluid for hemodynamic applications, J. Comput. Sci., 30 (2019), 79-89.
doi: 10.1016/j.jocs.2018.11.010. |
[19] |
M. Mu and J. Xu,
A two-grid method of a mixed Stokes–Darcy model for coupling fluid flow with porous media flow, SIAM J. Numer. Anal., 45 (2007), 1801-1813.
doi: 10.1137/050637820. |
[20] |
M. Mu and X. Zhu,
Decoupled schemes for a non-stationary mixed Stokes-Darcy model, Math. Comp., 79 (2010), 707-731.
doi: 10.1090/S0025-5718-09-02302-3. |
[21] |
A. Quarteroni and A. Valli, Domain Decomposition Methods for Partial Differential Equations, Numerical Mathematics and Scientific Computation, The Clarendon Press, Oxford University Press, New York, 1999. |
[22] |
M. Razzaq, H. Damanik, J. Hron, A. Ouazzi and S. Turek,
FEM multigrid techniques for fluid-structure interaction with application to hemodynamics, Appl. Numer. Math., 62 (2012), 1156-1170.
doi: 10.1016/j.apnum.2010.12.010. |
[23] |
B. Rivière,
Analysis of a discontinuous finite element method for the coupled Stokes and Darcy problems, J. Sci. Comput., 22/23 (2005), 479-500.
doi: 10.1007/s10915-004-4147-3. |
[24] |
B. Rivière and I. Yotov,
Locally conservative coupling of Stokes and Darcy flows, SIAM J. Numer. Anal., 42 (2005), 1959-1977.
doi: 10.1137/S0036142903427640. |
[25] |
R. K. Sinha and B. Deka,
Optimal error estimates for linear parabolic problems with discontinuous coefficients, SIAM J. Numer. Anal., 43 (2005), 733-749.
doi: 10.1137/040605357. |
[26] |
P. Sun and C. Wang,
Distributed Lagrange multiplier/fictitious domain finite element method for Stokes/parabolic interface problems with jump coefficients, Appl. Numer. Math., 152 (2020), 199-220.
doi: 10.1016/j.apnum.2019.12.009. |
[27] |
V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Springer Series in Computational Mathematics, 25, Springer-Verlag, Berlin, 2006.
doi: 10.1007/3-540-33122-0. |
[28] |
S. Turek and J. Hron, Proposal for numerical benchmarking of fluid-structure interaction between an elastic object and laminar incompressible flow, in Fluid Structure Interaction, Lect. Notes Comput. Sci. Eng., 53, Springer, Berlin, 2006,371–385.
doi: 10.1007/3-540-34596-5_15. |
[29] |
S. Turek, J. Hron, M. Razzaq, H. Wobker and M. Schäfer, Numerical benchmarking of fluid-structure interaction: A comparison of different discretization and solution approaches, In Fluid Structure Interaction II, Lect. Notes Comput. Sci. Eng., 73, Springer, Heidelberg, 2010,413–424.
doi: 10.1007/978-3-642-14206-2_15. |
[30] |
T. Wick,
Solving monolithic fluid-structure interaction problems in arbitrary Lagrangian Eulerian coordinates with the deal. II Library, Arch. Numer. Software, 1 (2013), 1-19.
doi: 10.11588/ans.2013.1.10305. |


h | Coupled scheme | Decoupled iRN scheme |
3.43107e-2 | 3.75366e-2 | |
9.20988e-3 | 1.13574e-2 | |
2.34374e-3 | 2.61407e-3 | |
5.88545e-4 | 5.77793e-4 |
h | Coupled scheme | Decoupled iRN scheme |
3.43107e-2 | 3.75366e-2 | |
9.20988e-3 | 1.13574e-2 | |
2.34374e-3 | 2.61407e-3 | |
5.88545e-4 | 5.77793e-4 |
h | Coupled scheme | Decoupled DN scheme | |
3.43107e-2 | 3.53891e-2 | ||
9.20988e-3 | 9.46749e-3 | ||
2.34374e-3 | 2.40475e-3 | ||
h | Decoupled iRR scheme | Decoupled RR scheme | |
3.53436e-2 | 3.24908e-2 | 3.80028e-2 | |
9.55398e-3 | 8.81404e-3 | 1.39637e-2 | |
2.25541e-3 | 2.26911e-3 | 5.08893e-3 |
h | Coupled scheme | Decoupled DN scheme | |
3.43107e-2 | 3.53891e-2 | ||
9.20988e-3 | 9.46749e-3 | ||
2.34374e-3 | 2.40475e-3 | ||
h | Decoupled iRR scheme | Decoupled RR scheme | |
3.53436e-2 | 3.24908e-2 | 3.80028e-2 | |
9.55398e-3 | 8.81404e-3 | 1.39637e-2 | |
2.25541e-3 | 2.26911e-3 | 5.08893e-3 |
h | Coupled scheme | Decoupled DN scheme | |
3.36012e-2 | |||
9.01136e-3 | |||
2.29268e-3 | |||
h | Decoupled iRR scheme | Decoupled RR scheme | |
3.24630e-2 | 3.18627e-2 | 3.72954e-2 | |
8.80706e-3 | 8.64307e-3 | 1.37006e-2 | |
2.27580e-3 | 2.26911e-3 | 5.00114e-3 |
h | Coupled scheme | Decoupled DN scheme | |
3.36012e-2 | |||
9.01136e-3 | |||
2.29268e-3 | |||
h | Decoupled iRR scheme | Decoupled RR scheme | |
3.24630e-2 | 3.18627e-2 | 3.72954e-2 | |
8.80706e-3 | 8.64307e-3 | 1.37006e-2 | |
2.27580e-3 | 2.26911e-3 | 5.00114e-3 |
h | Coupled scheme | Decoupled DN scheme | |
3.39101e-2 | 3.21780e-2 | ||
9.09826e-3 | 9.38034e-3 | ||
2.31506e-3 | 2.38207e-3 | ||
h | Decoupled iRR scheme | Decoupled RR scheme | |
3.23888e-2 | 3.21780e-2 | 3.76394e-2 | |
8.82536e-3 | 8.72945e-3 | 1.38221e-2 | |
2.28889e-3 | 2.24812e-3 | 5.04232e-3 |
h | Coupled scheme | Decoupled DN scheme | |
3.39101e-2 | 3.21780e-2 | ||
9.09826e-3 | 9.38034e-3 | ||
2.31506e-3 | 2.38207e-3 | ||
h | Decoupled iRR scheme | Decoupled RR scheme | |
3.23888e-2 | 3.21780e-2 | 3.76394e-2 | |
8.82536e-3 | 8.72945e-3 | 1.38221e-2 | |
2.28889e-3 | 2.24812e-3 | 5.04232e-3 |
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