Decoupling PDE computation with intrinsic or inertial Robin interface condition

Lian Zhang; Mingchao Cai; Mo Mu; Lian Zhang; Mingchao Cai; Mo Mu

doi:10.3934/era.2020102

Electronic Research Archive

2021, Volume 29, Issue 2: 2007-2028. doi: 10.3934/era.2020102

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Decoupling PDE computation with intrinsic or inertial Robin interface condition

Lian Zhang ^{1
,},
Mingchao Cai ^{2
,
,},
Mo Mu ^{1
,}

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

Received: 01 May 2020 Revised: 01 August 2020 Published: 23 September 2020
65N30, 65M55, 74F10

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.
- Multi-domain,
- multi-physics models,
- coupled PDEs,
- interface coupling conditions,
- decoupling,
- intrinsic or inertial type Robin condition,
- decoupled numerical methods
Citation: Lian Zhang, Mingchao Cai, Mo Mu. Decoupling PDE computation with intrinsic or inertial Robin interface condition[J]. Electronic Research Archive, 2021, 29(2): 2007-2028. doi: 10.3934/era.2020102

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Abstract

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]	Numerical analysis of the Navier–Stokes/Darcy coupling. Numer. Math. (2010) 115: 195-227.
[2]	Fluid–structure partitioned procedures based on Robin transmission conditions. J. Comput. Phys. (2008) 227: 7027-7051.
[3]	Scalable parallel methods for monolithic coupling in fluid–structure interaction with application to blood flow modeling. J. Comput. Phys. (2010) 229: 642-659.
[4]	Stability and convergence analysis of the extensions of the kinematically coupled scheme for the fluid-structure interaction. SIAM J. Numer. Anal. (2016) 54: 3032-3061.
[5]	Numerical solution to a mixed Navier–Stokes/Darcy model by the two-grid approach. SIAM J. Numer. Anal. (2009) 47: 3325-3338.
[6]	Preconditioning techniques for a mixed Stokes/Darcy model in porous media applications. J. Comput. Appl. Math. (2009) 233: 346-355.
[7]	Stability of the kinematically coupled $\beta$-scheme for fluid-structure interaction problems in hemodynamics. Int. J. Numer. Anal. Model. (2015) 12: 54-80.
[8]	Coupled Stokes-Darcy model with Beavers-Joseph interface boundary condition. Commun. Math. Sci. (2010) 8: 1-25.
[9]	Added-mass effect in the design of partitioned algorithms for fluid–structure problems. Comput. Methods Appl. Mech. Engrg. (2005) 194: 4506-4527.
[10]	Finite element methods and their convergence for elliptic and parabolic interface problems. Numer. Math. (1998) 79: 175-202.
[11]	Fluid–structure interaction simulation of aortic blood flow. Comput. & Fluids (2011) 43: 46-57.
[12]	Navier-Stokes/Darcy coupling: Modeling, analysis, and numerical approximation. Rev. Mat. Complut. (2009) 22: 315-426.
[13]	Robin–Robin domain decomposition methods for the Stokes–Darcy coupling. SIAM J. Numer. Anal. (2007) 45: 1246-1268.
[14]	Explicit Robin–Neumann schemes for the coupling of incompressible fluids with thin-walled structures. Comput. Methods Appl. Mech. Engrg. (2013) 267: 566-593.
[15]	Generalized Robin–Neumann explicit coupling schemes for incompressible fluid-structure interaction: Stability analysis and numerics. Internat. J. Numer. Methods Engrg. (2015) 101: 199-229.
[16]	Artificial added mass instabilities in sequential staggered coupling of nonlinear structures and incompressible viscous flows. Comput. Methods Appl. Mech. Engrg. (2007) 196: 1278-1293.
[17]	Mixed finite element analysis for an elliptic/mixed elliptic interface problem with jump coefficients. Procedia Comput. Sci. (2017) 108: 1913-1922.
[18]	Numerical simulation of an immersed rotating structure in fluid for hemodynamic applications. J. Comput. Sci. (2019) 30: 79-89.
[19]	A two-grid method of a mixed Stokes–Darcy model for coupling fluid flow with porous media flow. SIAM J. Numer. Anal. (2007) 45: 1801-1813.
[20]	Decoupled schemes for a non-stationary mixed Stokes-Darcy model. Math. Comp. (2010) 79: 707-731.
[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]	FEM multigrid techniques for fluid-structure interaction with application to hemodynamics. Appl. Numer. Math. (2012) 62: 1156-1170.
[23]	Analysis of a discontinuous finite element method for the coupled Stokes and Darcy problems. J. Sci. Comput. (2005) 22/23: 479-500.
[24]	Locally conservative coupling of Stokes and Darcy flows. SIAM J. Numer. Anal. (2005) 42: 1959-1977.
[25]	Optimal error estimates for linear parabolic problems with discontinuous coefficients. SIAM J. Numer. Anal. (2005) 43: 733-749.
[26]	Distributed Lagrange multiplier/fictitious domain finite element method for Stokes/parabolic interface problems with jump coefficients. Appl. Numer. Math. (2020) 152: 199-220.
[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]	Solving monolithic fluid-structure interaction problems in arbitrary Lagrangian Eulerian coordinates with the deal. II Library. Arch. Numer. Software (2013) 1: 1-19.

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